2. Magical equation for a contactification

The purpose of this section is to prove the following crucial lemma about a certain algebraic equation, which we call a magical equation. It is the boxed equation (2.1) below.

Lemma 2.1. Let $(\mathfrak {n}_{00},[\cdot ,\cdot ]_0)$ be a finite dimensional Lie algebra, and let $\rho :\mathfrak {n}_{00}\stackrel {\mathrm {hom}}{\to } \mathrm {End}(S)$ be its finite dimensional representation in a real vector space S of dimension s. In addition, let R be an r-dimensional real vector space, and $\tau :\mathfrak {n}_{00}\to \mathrm {End}(R)$ , be a linear map. Finally, let $\omega $ be a linear map $\omega :\bigwedge ^2S\to R$ , or what is the same, let $\omega \in \mathrm {Hom}(\bigwedge ^2S,R)$ .

Suppose now that the triple $(\rho ,\omega ,\tau )$ satisfies the following equation:

(2.1) $$ \begin{align}\boxed{ \omega\big(\rho(A)X,Y\big)+\omega\big(X,\rho(A)Y\big)=\tau(A)\,\omega(X,Y),} \end{align} $$

for all $A\in \mathfrak {n}_{00}$ and all $X,Y\in S$ . Then we have the following:

1. (1) The map $\tau $ satisfies

   $$ \begin{align*}\big(\,\,\tau([A,B]_0)-[\tau(A),\tau(B)]_{\mathrm{End}(R)}\,\,\big)\omega\,\,=\,\,0\quad\quad \forall\,\,A,B\in\mathfrak{n}_{00}.\end{align*} $$

2. (2) If the map $\tau :\mathfrak {n}_{00}\to \mathrm {End}(R)$ is a representation of $\mathfrak {n}_{00}$ , that is, if

   $$ \begin{align*}\tau([A,B]_0)=[\tau(A),\tau(B)]_{\mathrm{End}(R)},\end{align*} $$
   then the real vector space $\mathfrak {g}_0:=R\oplus S\oplus \mathfrak {n}_{00}$ is a graded Lie algebra
   $$ \begin{align*}\mathfrak{g}_0=\mathfrak{n}_{-2}\oplus\mathfrak{n}_{-1}\oplus\mathfrak{n}_{00},\end{align*} $$
   with the graded components
   $$ \begin{align*}\mathfrak{n}_{-2}=R,\quad \mathfrak{n}_{-1}=S,\quad\mathrm{with}\,\, \mathfrak{n}_{00}\,\,\mathrm{as\,\,the}\,\,0\,\,\mathrm{grade},\end{align*} $$
   and with the Lie bracket $[\cdot ,\cdot ]$ given by

   1. (a) if $X,Y\in \mathfrak {n}_{00}$ , then $[X,Y]=[X,Y]_0$ ,

   2. (b) if $A\in \mathfrak {n}_{00}$ , $X\in \mathfrak {n}_{-1}$ , then $[A,X]=\rho (A)X$ ,

   3. (c) if $A\in \mathfrak {n}_{00}$ , $X\in \mathfrak {n}_{-2}$ , then $[A,X]=\tau (A)X$ ,

   4. (d) $[\mathfrak {n}_{-1},\mathfrak {n}_{-2}]=[\mathfrak {n}_{-2},\mathfrak {n}_{-2}]=\{0\}$ ,

   5. (e) and, if $X,Y\in \mathfrak {n}_{-1}$ , then $[X,Y]=\omega (X,Y)$ .

3. (3) Moreover, in the case (2), the Lie subalgebra

   
   of $\mathfrak {g}_0$ is a 2-step graded Lie algebra, and the algebra $\mathfrak {n}_{00}$ is a Lie subalgebra of the Lie algebra
   
   of all derivations of preserving its strata $\mathfrak {n}_{-1}$ and $\mathfrak {n}_{-2}$ .

Remark 2.2. Note that, in the respective bases $\{ f_\mu \}_{\mu =1}^s$ in S and $\{e_i\}_{i=1}^r$ in R, the equation (2.1) is

(2.2) $$ \begin{align}\boxed{ \rho(A)^\alpha{}_\mu\,\,\omega^i{}_{\alpha\nu}+\rho(A)^\alpha{}_\nu\,\,\omega^i{}_{\mu\alpha}\,\,=\,\,\tau(A)^i{}_j\,\,\omega^i{}_{\mu\nu}}\end{align} $$

for all $A\in \mathfrak {n}_{00}$ , all $i=1,2,\dots ,r$ and all $\mu ,\nu =1,2,\dots ,s$ . On this basis, the condition (1) is

$$ \begin{align*}\big(\,\,\tau([A,B]_0)-[\tau(A),\tau(B)]_{\mathrm{End}(R)}\,\,\big)^i{}_j\,\,\omega^j{}_{\mu\nu}\,\,=\,\,0\end{align*} $$

for all $i=1,2,\dots ,r,\,\mu ,\nu =1,2,\dots s$ , and $A,B\in \mathfrak {n}_{00}$ .

Proof of the lemma

The proof of part (1) is a pure calculation using the equation (2.1). We first rewrite it in the shorthand notation as

$$ \begin{align*}\rho(A)\omega+\omega\rho(A)^T=\tau(A)\omega, \quad \forall A\in\mathfrak{n}_{00}.\end{align*} $$

Then we have

$$ \begin{align*}\begin{aligned} \tau([A,B]_0)\omega&=\rho([A,B]_0)\omega+\omega\rho([A,B]_0)^T\\ &=\rho(A)\rho(B)\omega-\rho(B)\rho(A)\omega+\omega\rho(B)^T\rho(A)^T-\omega\rho(A)^T\rho(B)^T\\ &=\rho(A)\Big(\tau(B)\omega-\omega\rho(B)^T\Big)-\rho(B)\Big(\tau(A)\omega-\omega\rho(A)^T\Big)\\&\quad +\Big(\tau(B)\omega-\rho(B)\omega\Big)\rho(A)^T-\Big(\tau(A)\omega-\rho(A)\omega\Big)\rho(B)^T\\ &=\rho(A)\Big(\tau(B)\omega\Big)-\rho(B)\Big(\tau(A)\omega\Big)+\Big(\tau(B)\omega\Big)\rho(A)^T-\Big(\tau(A)\omega\Big)\rho(B)^T\\ &= \tau(A)\tau(B)\omega-\tau(B)\omega\rho(A)^T-\Big(\tau(B)\tau(A)\omega-\tau(A)\omega\rho(B)^T\Big)\\ &\quad +\tau(B)\omega\rho(A)^T-\tau(A)\omega\rho(B)^T=\tau(A)\tau(B)\omega-\tau(B)\tau(A)\omega\\ &=([\tau(A,\tau(B)]_{\mathrm{End}(R)})\omega, \end{aligned} \end{align*} $$

which proves part (1).

The proof of parts (2) and (3) is as follows:

We need to check the Jacobi identity for the bracket $[\cdot ,\cdot ]$ .

We first consider the representation

defined by

$$ \begin{align*}\sigma(A)(Y\oplus X)=\tau(A)Y\oplus\rho(A)X, \quad \forall A\in\mathfrak{n}_{00},\,\, X\in\mathfrak{n}_{-1},\,\, Y\in\mathfrak{n}_{-2}.\end{align*} $$

We then prove that the representation $\sigma $ is a strata preserving derivation in . This is implied by the definitions (a)–(e) of the bracket, and the fundamental equation (2.1) as follows:

The strata preserving property of $\sigma $ , $\sigma (\mathfrak {n}_{-i})\subset \mathfrak {n}_{-i}$ , $i=1,2$ , is obvious by the definitions of $\rho $ and $\tau $ . However, we need to check that $\sigma $ is a derivation – that is, that

(2.3) $$ \begin{align}\sigma (A)[X,Y] =[\sigma(A)X,Y]+[X,\sigma(A)Y]\end{align} $$

for all $A\in \mathfrak {n}_{00}$ and for all . Because of the strata preserving property of $\sigma $ , which we have just established, and because of the point (d) of the definition of the bracket, the equation (2.3) is satisfied when both X and Y are in $\mathfrak {n}_{-2}$ , or when X is in $\mathfrak {n}_{-1}$ and Y is $\mathfrak {n}_{-2}$ . The only thing to be checked is if (2.3) is also valid when both X and Y belong to $\mathfrak {n}_{-1}$ . But this just follows directly from (2.1) since if $ X,Y\in \mathfrak {n}_{-1}$ , then

$$ \begin{align*}\begin{aligned}\sigma(A)[X,Y]&=\sigma(A)\omega(X,Y)=\tau(A)\omega(X,Y)=\\ &\omega(\rho(A)X,Y)+\omega(X,\rho(A)Y)=[\rho(A)X,Y]+[X,\rho(A)Y]=\\ &[\sigma(A)X,Y]+[X,\sigma(A)Y],\quad \forall A\in\mathfrak{n}_{00}.\end{aligned}\end{align*} $$

Now we return to checking the Jacobi identity for the bracket $[\cdot ,\cdot ]$ in $\mathfrak {g}_0$ :

On elements of the form $A,B\in \mathfrak {n}_{00}$ , , by (b)-(c), we have

$$ \begin{align*}[[A,B],Z]+[[Z,A],B]+[[B,Z],A]=\Big(\sigma([A,B])-[\sigma(A),\sigma(B)]\Big)Z,\end{align*} $$

which vanishes due to the representation property of $\sigma $ . However, on elements $A\in \mathfrak {n}_{00}$ and , we have

$$ \begin{align*}[[A,Z_1],Z_2]+[[Z_2,A],Z_1]+[[Z_1,Z_2],A]=[\sigma(A)Z_1,Z_2]+[Z_1,\sigma(A)Z_2]-\sigma(A)[Z_1,Z_2],\end{align*} $$

which is again zero, on the ground of the derivation property 2.3 of $\sigma $ . Obviously, the bracket satisfies the Jacobi identity when it is restricted to $\mathfrak {n}_{00}$ ; it is the Lie bracket $[\cdot ,\cdot ]_o$ of the Lie algebra $\mathfrak {n}_{00}$ . Finally, property (2) implies that $[[Z_1,Z_2],Z_3]=0$ for all $Z_1,Z_2,Z_3$ in ; hence, the Jacobi identity is trivially satisfied for $[\cdot ,\cdot ]$ , when it is restricted to .

In the following, we will use the map $\omega \in \mathrm {Hom}(\bigwedge ^2S,R)$ satisfying the magical equation (2.1) to construct contactifications with nontrivial symmetry algebras $\mathfrak {g}$ . The setting will include Cartan’s contactification with symmetry $\mathbf {F}_4$ mentioned in the Helgason’s quote. For this, however, we need few preparations.

3. Two-step filtered manifolds

A 2-step filtered structure on an $(s+r)$ -dimensional manifold M is a pair $(M,{\mathcal D})$ , in which $\mathcal D$ is a vector distribution of rank s on M, such that it is bracket generating in the quickest possible way. This means that its derived distribution ${\mathcal D}_{-2}:=[{\mathcal D}_{-1},{\mathcal D}_{-1}]$ , with ${\mathcal D}_{-1}={\mathcal D}$ , is such that

$$ \begin{align*}{\mathcal D}_{-2}=\mathrm{T}M.\end{align*} $$

It provides the simplest nontrivial filtration

$$ \begin{align*}\mathrm{T}M={\mathcal D}_{-2}\supset{\mathcal D}_{-1}\end{align*} $$

of the tangent bundle $\mathrm {T}M$ .

A (local) authomorphism of a 2-step filtered manifold $(M,{\mathcal D})$ is a (local) diffeomorphism $\phi :M\to M$ such that $\phi _*{\mathcal D}\subset {\mathcal D}$ . Since authomorphisms can be composed and have inverses, they form a group G of (local) authomorphisms of $(M,{\mathcal D})$ , also called a group of (local) symmetries of $\mathcal D$ . Infinitesimally, the Lie group of authomorphisms defines the Lie algebra $\mathfrak {aut}({\mathcal D})$ of symmetries, which is the real span of all vector fields X on M such that $[X,Y]\subset {\mathcal D}$ for all $Y\in {\mathcal D}$ .

Among all the 2-step filtered manifolds $(M,{\mathcal D})$ particularly simple are those which can be realised on a group manifold of a 2-step nilpotent Lie group. These are related to the notion of the nilpotent approximation of a pair $(M,{\mathcal D})$ . This is defined as follows:

At every point $x\in M$ equipped with a 2-step filtration ${\mathcal D}_{-2}\supset {\mathcal D}_{-1}$ , we have well-defined vector spaces $n_{-1}(x)={\mathcal D}_{-1}(x)$ and $n_{-2}(x)={\mathcal D}_{-2}(x)/{\mathcal D}_{-1}(x)$ , which define a vector space

$$ \begin{align*}\mathfrak{n}(x)=\mathfrak{n}_{-2}(x)\oplus\mathfrak{n}_{-1}(x).\end{align*} $$

This vector space is naturally a Lie algebra, with a Lie bracket induced form the Lie bracket of vector fields in $\mathrm {T}M$ . Due to the 2-step property of the filtration defined by $\mathcal D$ , this Lie algebra is 2-step nilpotent,

$$ \begin{align*}[\mathfrak{n}_{-1}(x),\mathfrak{n}_{-1}(x)]=\mathfrak{n}_{-2}(x)\quad\&\quad[\mathfrak{n}_{-1}(x),\mathfrak{n}_{-2}(x)]=\{0\}.\end{align*} $$

This 2-step nilpotent Lie algebra is a local invariant of the structure $(M,{\mathcal D})$ , and it is called a nilpotent approximation of the structure $(M,{\mathcal D})$ at $x\in M$ .

This enables for defining a class of particularly simple examples of 2-step filtered structures:

Consider a 2-step nilpotent Lie algebra $\mathfrak {n}=\mathfrak {n}_{-2}\oplus \mathfrak {n}_{-1}$ , and let M be a Lie group, whose Lie algebra is $\mathfrak {n}$ . The Lie algebra $\mathfrak {n}_M$ of left invariant vector fields on M is isomorphic to $\mathfrak {n}$ and mirrors its gradation, $\mathfrak {n}_M={\mathfrak {n}_M}_{-2}\oplus {\mathfrak {n}_M}_{-1}$ . Now, taking all linear combinations with smooth functions coefficients of all vector fields from the graded component ${\mathfrak {n}_M}_{-1}$ of $\mathfrak {n}_M$ , one defines a vector distribution ${\mathcal D}=\mathrm {Span}_{{\mathcal {F}}(M)}(\mathfrak {n}_M)$ on M. The so constructed filtered structure $(M,{\mathcal D})$ is obviously 2-step graded and is the simplest filtered structure with nilpotent approximation being equal to $\mathfrak {n}$ everywhere. We call this $(M,{\mathcal D})$ structure the flat model for all the 2-step filtered structures having the same constant nilpotent approximation $\mathfrak {n}$ .

It is remarkable that the largest possible symmetry of all 2-step filtered structures $(M,{\mathcal D})$ is precisely the symmetry of the flat model. As such, it is algebraically determined by the nilpotent approximation $\mathfrak {n}$ . This is the result of Noboru Tanaka [Reference Tanaka18]. To describe it, we recall the notion of Tanaka prolongation.

Definition 3.1. The Tanaka prolongation of a 2-step nilpotent Lie algebra $\mathfrak {n}$ is a graded Lie algebra $\mathfrak {g}(\mathfrak {n})$ given by a direct sum

(3.1) $$ \begin{align} \mathfrak{g}(\mathfrak{n})=\mathfrak{n}\oplus\mathfrak{n}_0\oplus\mathfrak{n}_1\oplus\dots\oplus\mathfrak{n}_j\oplus\cdots,\end{align} $$

with

(3.2) $$ \begin{align}\mathfrak{n}_k=\Big\{\bigoplus_{j<0}\mathfrak{n}_{k+j}\otimes\mathfrak{n}_j^*\ni A\,\,\mathrm{s.t.}\,\,A[X,Y]=[AX,Y]+[X,AY]\Big\}\end{align} $$

for each $k\geq 0$ .

Furthermore, for each $j\geq 0$ , the Lie algebra

$$ \begin{align*}\mathfrak{g}_j(\mathfrak{n})=\mathfrak{n}\oplus\mathfrak{n}_0\oplus\mathfrak{n}_1\oplus\dots\oplus\mathfrak{n}_j\end{align*} $$

is called the Tanaka prolongations of $\mathfrak {n}$ up to $j^{th}$ order.

Setting $[A,X]=AX$ for all $A\in \mathfrak {n}_k$ with $k\geq 0$ and for all $X\in \mathfrak {n}$ makes the condition in (3.2) into the Jacobi identity. Moreover, if $A\in \mathfrak {n}_k$ and $B\in \mathfrak {n}_l$ , $k,l\geq 0$ , then their commutator $[A,B]\in \mathfrak {n}_{k+l}$ is defined on elements $X\in \mathfrak {n}$ inductively, according to the Jacobi identity. By this, we mean that it should satisfy

$$ \begin{align*}[A,B]X=[A,BX]-[B,AX],\end{align*} $$

which is sufficient enough to define $[A,B]$ .

Remark 3.2. Note, in particular, that $\mathfrak {n}_0$ is the Lie algebra of all derivations of $\mathfrak {n}$ preserving the two strata $\mathfrak {n}_{-1}$ and $\mathfrak {n}_{-2}$ of the direct sum $\mathfrak {n}=\mathfrak {n}_{-2}\oplus \mathfrak {n}_{-1}$ :

$$ \begin{align*}\mathfrak{n}_0=\big\{\,\mathrm{Der}(\mathfrak{n})\ni D\,\,\mathrm{s.t.}\,\,D\mathfrak{n}_j\subset\mathfrak{n}_j\,\,\mathrm{for}\,\, j=-1,-2\,\big\}.\end{align*} $$

Although the Tanaka prolongation of a nilpotent Lie algebra $\mathfrak {n}$ is in general infinite, in this paper, we will be interested in situations when the Tanaka prolongation

$$ \begin{align*}\mathfrak{g}=\mathfrak{g}(\mathfrak{n})\end{align*} $$

of the $2$ -step nilpotent part

$$ \begin{align*}\mathfrak{n}=\mathfrak{n}_{-2}\oplus\mathfrak{n}_{-1}\end{align*} $$

is finite and symmetric, in the sense

$$ \begin{align*}\mathfrak{g}(\mathfrak{n})=\mathfrak{n}_{-2}\oplus\mathfrak{n}_{-1}\oplus\mathfrak{n}_0\oplus\mathfrak{n}_1\oplus\mathfrak{n}_2,\end{align*} $$

with

$$ \begin{align*}\dim(\mathfrak{n}_{-k})=\dim(\mathfrak{n}_k), \quad k=1,2.\end{align*} $$

Such situations are possible, and in them the so defined Lie algebra $\mathfrak {g}(\mathfrak {n})$ is simple. In such a case, the Tanaka prolongation $\mathfrak {g}(\mathfrak {n})$ is graded, and the subalgebra

$$ \begin{align*}\mathfrak{p}=\mathfrak{n}_0\oplus\mathfrak{n}_1\oplus\mathfrak{n}_2\end{align*} $$

in such $\mathfrak {g}(\mathfrak {n})$ is parabolic. Moreover, the Lie algebra

$$ \begin{align*}\mathfrak{p}_{opp}=\mathfrak{n}_{-2}\oplus\mathfrak{n}_{-1}\oplus\mathfrak{n}_0\end{align*} $$

is also a parabolic subalgebra of this simple $\mathfrak {g}(\mathfrak {n})$ . It is isomorphic to $\mathfrak {p}$ , $\mathfrak {p}\simeq \mathfrak {p}_{opp}$ .

Regardless of whether $\mathfrak {g}(\mathfrak {n})$ is finite or not, we have the following general theorem, which is a specialisation of a remarkable theorem by Noboru Tanaka [Reference Tanaka18]:

Using the information about the Tanaka prolongation of a nilpotent Lie algebra $\mathfrak {n}$ , we can enlarge our Lemma 2.1 by changing its point (3) into the following more complete form:

Remark 3.6. The term ‘... $\mathfrak {n}_{00}$ is a Lie subalgebra of the Tanaka prolongation up to $0^{th}$ order of the Lie algebra ..’ in the above lemma means that $\mathfrak {n}_{00}$ , although nontrivial, is in general only a subalgebra of the

which is the full $0$ graded component of the Tanaka prolongation of . So for applications, it is reasonable to choose $\mathfrak {n}_{00}$ as large as possible.

4. Construction of contactifications with nice symmetries

Consider a Lie algebra $(\mathfrak {n}_{00},[\cdot ,\cdot ]_0)$ and its two real representations $(\rho ,S)$ , $(\tau ,R)$ , in the respective real s- and r-dimensional vector spaces S and R. Let $S=\mathbb {R}^s$ , $R=\mathbb {R}^r$ , and let $\{ f_\mu \}_{\mu =1}^s$ and $\{e_i\}_{i=1}^r$ be respective bases in S and in R. Let $\{f^\mu \}_{\mu =1}^s$ be a basis in the vector space $S^*$ dual to the basis $\{ f_\mu \}_{\mu =1}^s$ , . To be in a situation of Lemma 2.1, we also assume that we have the homomorphism $\omega \in \mathrm {Hom}(\bigwedge ^2 S,R)$ satisfying the magical equation (2.1).

Then the map $\omega $ is

$$ \begin{align*}\omega=\tfrac12\omega^i{}_{\mu\nu}e_i\otimes f^\mu\wedge f^\nu,\end{align*} $$

and it defines the coefficients $\omega ^i{}_{\mu \nu }$ , $i=1,\dots ,r$ , $\mu ,\nu =1,2,\dots s$ , which satisfy $\omega ^i{}_{\mu \nu }=-\omega ^i{}_{\nu \mu }$ .

Now, consider an s-dimensional manifold, which is an open set N of $\mathbb {R}^s$ , $N\subset \mathbb {R}^s$ , with coordinates $(x^\mu )_{\mu =1}^r$ . Then we have r two-forms $(\omega ^i)_{i=1}^r$ on N, defined by

$$ \begin{align*}\omega^i=\tfrac12\omega^i{}_{\mu\nu}\mathrm{d} x^\mu\wedge\mathrm{d} x^\nu.\end{align*} $$

This produces an $(N,\mathrm {d}{\mathcal D}^\perp )$ structure on N, with

$$ \begin{align*}\mathrm{d}{\mathcal D}^\perp=\mathrm{Span}_{\mathbb{R}}(\omega^1,\dots,\omega^r).\end{align*} $$

We contactify it. For this, we take a local $M=\mathbb {R}^r\times N$ , with coordinates $\big (u^i,x^\mu \big )\phantom {}_{i=1}^r\phantom {}_{\mu =1}^s$ , and define the ‘contact forms’ on M by

$$ \begin{align*}\lambda^i=\mathrm{d} u^i+\omega^i{}_{\mu\nu}x^\mu\mathrm{d} x^\nu.\end{align*} $$

Because of Lemmas 2.1 and 3.5, the distribution ${\mathcal D}$ on M defined by this contactification as in Definition 1.1 equips M with a 2-step filtered structure having ${\mathcal D}_{-1}=\mathcal D$ . This has rank s. Now using Lemmas 2.1 and 3.5, and Tanaka’s Theorem 3.3, we get the following corollary.

Corollary 4.1. Let $M=\mathbb {R}^r\times \mathbb {R}^s$ and let

$$ \begin{align*}\lambda^i=\mathrm{d} u^i+\omega^i{}_{\mu\nu}x^\mu\mathrm{d} x^\nu, \quad i=1,\dots r,\end{align*} $$

with $\omega $ being a solution of the magical equation 2.1 such that $\mathrm {Im}(\omega )=R$ . Consider the distribution structure $(M,{\mathcal D})$ with a rank r distribution

on M. Then the Lie algebra of authomorphism $\mathfrak {aut}({\mathcal D})$ of $(M,{\mathcal D})$ is isomorphic to the Tanaka prolongation of the 2-step nilpotent Lie algebra defined in point (3) of Lemma 2.1 or 3.5. The Lie algebra $\mathfrak {g}_0=R\oplus S\oplus \mathfrak {n}_{00}$ is nontrivially contained in the Tanaka prolongation up to the $0^{th}$ order of , with $\{0\}\neq \mathfrak {n}_{00}\subset \mathfrak {n}_0$ , and as such is a subalgebra of the algebra of $\mathfrak {aut}({\mathcal D})$ .

5. Majorana spinor representations of $\mathfrak {so}(p,q)$

In this section, we will explain how to construct the real spin representations of the Lie algebras $\mathfrak {so}(p,q)$ , in cases when $p=n$ , $q=n-1$ , or $p=q=n$ , $n=1,2,\dots n$ . We will also give a construction of these representations for $\mathfrak {so}(0,n)$ . We emphasise that we are only interested in real spin representations. They share a general name of Majorana representations. Our presentation of this material is adapted from [Reference Trautman, Naber, Tsou and Françoise19].

We will need Pauli matrices

(5.1) $$ \begin{align} \sigma_x=\begin{pmatrix} 0&1\\1&0\end{pmatrix},\quad \epsilon=-i\sigma_y=\begin{pmatrix} 0&-1\\1&0\end{pmatrix},\quad \sigma_z=\begin{pmatrix} 1&0\\0&-1\end{pmatrix} \end{align} $$

and the $2\times 2$ identity matrix

(5.2) $$ \begin{align} I=\begin{pmatrix} 1&0\\0&1\end{pmatrix}.\end{align} $$

We have the following identities:

(5.3) $$ \begin{align}\begin{aligned} &\sigma_x^2=\sigma_z^2=-\epsilon^2=I\\ &\sigma_x\epsilon=-\epsilon\sigma_x=\sigma_z,\quad\sigma_z\sigma_x=-\sigma_x\sigma_z=-\epsilon,\quad \epsilon\sigma_z=-\sigma_z\epsilon=\sigma_x. \end{aligned} \end{align} $$

Now we quote [Reference Trautman, Naber, Tsou and Françoise19]:

With this notation, restricting to low dimensions $p+q=4,5,6$ and $7$ , the real representations of the Clifford algebra ${\mathcal C}\ell (0,p+q)$ are all in dimension $s=8$ , and are generated by the $p+q$ matrices $\rho _1,\dots ,\rho _{(p+q)}$ given by:

(5.4) $$ \begin{align}\begin{aligned} \rho_1&=\sigma_z\otimes I\otimes \epsilon\\ \rho_2&=\sigma_z\otimes \epsilon\otimes \sigma_x\\ \rho_3&=\sigma_z\otimes \epsilon\otimes \sigma_z\\ \rho_4&=\sigma_x\otimes \epsilon \otimes I\\ \rho_5&=\sigma_x\otimes \sigma_x\otimes \epsilon\\ \rho_6&=\sigma_x\otimes \sigma_z\otimes \epsilon\\ \rho_7&=\epsilon\otimes I\otimes I. \end{aligned}\end{align} $$

The eight matrices $\theta _\mu =\sigma _x\otimes \rho _\mu $ , $\mu =1,\dots , 7$ and $\theta _8=\epsilon \otimes I \otimes I\otimes I$ , give the real representation of ${\mathcal C}\ell (0,8)$ in $S=\mathbb {R}^{16}$ . Dropping the first factor in $\rho _1,\rho _2,\rho _3$ one obtains the matrices generating a representation of ${\mathcal C}\ell (0,3)$ in $S=\mathbb {R}^4$ , etc.

Majorana representations of $\mathfrak {so}(n-1,n)$ in dimension $s=2^{n-1}$ are called Pauli representations, and Majorana representations of $\mathfrak {so}(n,n)$ in dimension $s=2^n$ are called Dirac representations.

To construct them, we need generalisations of the Pauli $\sigma $ matrices and Dirac $\gamma $ matrices. The construction of those is inductive.

It starts with $p+q=1$ with one matrix $\sigma _1=1$ , and for every $n=1,2,\dots $ , it alternates between $p+q=2n-1$ of Pauli matrices $\sigma _\mu $ , $\mu =1,\dots , 2n-1$ , and $p+q=2n$ of Dirac matrices $\gamma _\mu $ , $\mu =1,\dots , 2n$ .

Again quoting Trautman [Reference Trautman, Naber, Tsou and Françoise19] we have the following:

1. (1) In dimension $p+q=1$ , put $\sigma _1=1$ .

2. (2) Given $2^{n-1}\times 2^{n-1}$ matrices $\sigma _\mu $ , $\mu =1,\dots ,2n-1$ , define

   $$ \begin{align*}\gamma_\mu=\begin{pmatrix} 0&\sigma_\mu\\\sigma_\mu&0\end{pmatrix}\,\,\mathrm{for}\,\,\mu=1,\dots,2n-1,\end{align*} $$
   and
   $$ \begin{align*}\gamma_{2n}=\begin{pmatrix} 0&-I\\I&0\end{pmatrix},\end{align*} $$
   where I is the identity $2^{n-1}\times 2^{n-1}$ matrix.

3. (3) Given $2^n\times 2^n$ matrices $\gamma _\mu $ , $\mu =1,\dots ,2n$ , define $\sigma _\mu =\gamma _\mu $ for $\mu =1,\dots , 2n$ and $\sigma _{2n+1}=\gamma _1\dots \gamma _{2n}$ , so that for $n>0$ ,

   $$ \begin{align*}\sigma_{2n+1}=\begin{pmatrix} I&0\\0&-I\end{pmatrix}.\end{align*} $$

In every dimension $p+q=2n-1$ , $n\geq 1$ , the Pauli matrices $\sigma _\mu $ , $\mu =1,\dots ,2n-1$ satisfy

$$ \begin{align*}\sigma_\mu\sigma_\nu+\sigma_\nu\sigma_\mu=2g_{\mu\nu}\underbrace{\big(I\otimes\dots\otimes I\big)}_{n-1\,\, \mathrm{times}},\end{align*} $$

where the $(2n-1)\times (2n-1)$ symmetric matrix $(g_{\mu \nu })$ is diagonal and has the following diagonal elements:

$$ \begin{align*}(g_{\mu\nu})=\mathrm{diag}\underbrace{(1,-1,\dots,-1,1)}_{(2n-1)\,\, \mathrm{times}}.\end{align*} $$

Likewise, in every dimension $p+q=2n$ , $n\geq 1$ , the Dirac matrices $\gamma _\mu $ , $\mu =1,\dots ,2n$ satisfy

$$ \begin{align*}\gamma_\mu\gamma_\nu+\gamma_\nu\gamma_\mu=2g_{\mu\nu}\underbrace{\big(I\otimes\dots\otimes I\big)}_{n\,\, \mathrm{times}},\end{align*} $$

where the $(2n)\times (2n)$ symmetric matrix $(g_{\mu \nu })$ is diagonal and has the following diagonal elements:

$$ \begin{align*}(g_{\mu\nu})=\mathrm{diag}\underbrace{(1,-1,\dots,1,-1)}_{2n\,\, \mathrm{times}}.\end{align*} $$

Therefore, for each $n=1,2,\dots $ , the set $\{\sigma _\mu \}_{\mu =1}^{2n-1}$ of Pauli matrices generates the elements of a real $2^{n-1}$ -dimensional representation of the Clifford algebra ${\mathcal C}\ell (n-1,n)$ , and the set $\{\gamma _\mu \}_{\mu =1}^{2n}$ of Dirac matrices generates the elements of a real $2^{n}$ -dimensional representation of the Clifford algebra ${\mathcal C}\ell (n,n)$ .

Then, in turn, these real Clifford algebras representations can be further used to define the real spin representations of the Lie algebras $\mathfrak {so}(p+q,0)$ , $\mathfrak {so}(n-1,n)$ and $\mathfrak {so}(n,n)$ as follows. One obtains all the generators of the spin representation of $\mathfrak {so}(g)$ by spanning it by all the elements of the form

• • $\tfrac 12\rho _\mu \rho _\nu $ , with $1\leq \mu <\nu \leq (p+q)$ , in the case of $\mathfrak {so}(p+q,0)$ , $p+q=3,5,6,7$ ;

• • $\tfrac 12\theta _\mu \theta _\nu $ , with $1\leq \mu <\nu \leq 8$ , in the case of $\mathfrak {so}(8,0)$ ;

• • $\tfrac 12\sigma _\mu \sigma _\nu $ , with $1\leq \mu <\nu \leq (p+q)=2n-1$ , in the case of $\mathfrak {so}(n-1,n)$ ;

• • $\tfrac 12\gamma _\mu \gamma _\nu $ , with $1\leq \mu <\nu \leq (p+q)=2n$ , in the case of $\mathfrak {so}(n,n)$ .

For further details, consult [Reference Trautman, Naber, Tsou and Françoise19].

We will use all this information in next sections when we create examples.

6. Application: Obtaining the flat model for (3,6) distributions

Let $(\rho ,S)$ be the defining representation of $\mathfrak {so}(3)$ in $S=\mathbb {R}^3$ . It can be generated by

(6.1) $$ \begin{align}\rho(A_1)=\begin{pmatrix} 0&0&-1\\0&0&0\\1&0&0\end{pmatrix},\quad\rho(A_2)=\begin{pmatrix} 0&1&0\\-1&0&0\\0&0&0\end{pmatrix},\quad\rho(A_1)=\begin{pmatrix} 0&0&0\\0&0&-1\\0&1&0\end{pmatrix}.\end{align} $$

And let $(\tau ,R)$ be an equivalent 3-dimensional representation of $\mathfrak {so}(3)$ given by

(6.2) $$ \begin{align} \tau(A_1)=\begin{pmatrix} 0&0&-1\\0&0&0\\1&0&0\end{pmatrix},\quad\tau(A_2)=\begin{pmatrix} 0&-1&0\\1&0&0\\0&0&0\end{pmatrix},\quad\tau(A_1)=\begin{pmatrix} 0&0&0\\0&0&1\\0&-1&0\end{pmatrix}.\end{align} $$

We claim that for these two representations of $\mathfrak {so}(3)$ , in the standard bases in $S=\mathbb {R}^3$ , $R=\mathbb {R}^3$ , the magical equation (2.2) has the following solution:

$$ \begin{align*}\omega^1{}_{\mu\nu}=\begin{pmatrix} 0&0&0\\0&0&-1\\0&1&0\end{pmatrix},\quad\omega^2{}_{\mu\nu}=\begin{pmatrix} 0&0&-1\\0&0&0\\1&0&0\end{pmatrix},\quad\omega^3{}_{\mu\nu}=\begin{pmatrix} 0&-1&0\\1&0&0\\0&0&0\end{pmatrix}.\end{align*} $$

Now using this solution $(\rho ,\tau ,\omega )$ of the magical equation (2.1), we use the Corollary 4.1 with $\lambda ^i=\mathrm {d} u^u+\omega ^i{}_{\mu \nu }x^\mu \mathrm {d} x^\nu $ and obtain the following theorem.

Theorem 6.1. Let $M=\mathbb {R}^6$ with coordinates $(u^1,u^2,u^3,x^1,x^2,x^3)$ and consider three 1-forms

$$ \begin{align*}\begin{aligned} \lambda^1& =\mathrm{d} u^1+x^2\mathrm{d} x^3\\ \lambda^2& =\mathrm{d} u^2+x^1\mathrm{d} x^3\\ \lambda^3& =\mathrm{d} u^3+x^1\mathrm{d} x^2 \end{aligned}\end{align*} $$

on M. Then the rank 3 distribution ${\mathcal D}$ on M defined by has its Lie algebra of infinitesimal symmetries $\mathfrak {aut}{\mathcal D}$ isomorphic to the Tanaka prolongation of , where $(\rho ,S=\mathbb {R}^3)$ and $(\tau ,R=\mathbb {R}^3)$ are the respective representations (6.1), (6.2) of $\mathfrak {n}_{00}=\mathfrak {so}(3)$ .

The symmetry algebra $\mathfrak {aut}({\mathcal D})$ is isomorphic to the simple graded Lie algebra $\mathfrak {so}(4,3)$ ,

$$ \begin{align*}\mathfrak{aut}({\mathcal D})=\mathfrak{so}(4,3),\end{align*} $$

with the following gradation:

$$ \begin{align*}\mathfrak{aut}({\mathcal D})=\mathfrak{n}_{-2}\oplus\mathfrak{n}_{-1}\oplus\mathfrak{n}_0\oplus\mathfrak{n}_1\oplus\mathfrak{n}_2,\end{align*} $$

with $\mathfrak {n}_{-2}=R$ , $\mathfrak {n}_{-1}=S$ ,

$$ \begin{align*}\mathfrak{n}_0=\mathfrak{gl}(3,\mathbb{R})\supset\mathfrak{n}_{00}\end{align*} $$

$\mathfrak {n}_{1}=S^*$ , $\mathfrak {n}_{2}=R^*$ , which is inherited from the distribution structure $(M,{\mathcal D})$ . The duality signs $*$ at $R^*$ and $S^*$ above are with respect to the Killing form in $\mathfrak {so}(4,3)$ .

The contactification $(M,{\mathcal D})$ is locally a flat model for the parabolic geometry of type $\big (\mathbf {Spin}(4,3),P\big )$ related to the following crossed Satake diagram: .

7. Application: Obtaining Biquard’s 7-dimensional flat quaternionic contact manifold via contactification using spin representations of $\mathfrak {so}(1,2)$ and $\mathfrak {so}(3,0)$

According to Trautman’s procedure, [Reference Trautman, Naber, Tsou and Françoise19] there is a real representation of ${\mathcal C}\ell (0,3)$ in $\mathbb {R}^4$ . There also is an analogous representation of ${\mathcal C}\ell (1,2)$ . Both of them are generated by the $\sigma $ matrices

$$ \begin{align*}\sigma_1=\begin{pmatrix} 0&-1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0\end{pmatrix},\quad\sigma_2=\begin{pmatrix} 0&0&0&-\varepsilon\\0&0&-\varepsilon&0\\0&1&0&0\\1&0&0&0\end{pmatrix},\quad\sigma_3=\begin{pmatrix} 0&0&-\varepsilon&0\\0&0&0&\varepsilon\\1&0&0&0\\0&-1&0&0\end{pmatrix},\end{align*} $$

where

$$ \begin{align*}\varepsilon=1\quad \mathrm{for}\quad {\mathcal C}\ell(0,3),\end{align*} $$

and

$$ \begin{align*}\varepsilon=-1\quad\mathrm{for}\quad{\mathcal C}\ell(2,1).\end{align*} $$

One can check that these matricesFootnote ¹ satisfy the (representation of) Clifford algebra relations:

$$ \begin{align*}\sigma_\mu\sigma_\nu+\sigma_\nu\sigma_\mu=2g_{\mu\nu}\big(I\otimes I)\end{align*} $$

with all $g_{\mu \nu }$ being zero, except $g_{11}=-1$ , $g_{22}=g_{33}=-\varepsilon $ .

This leads to the following spinorial representation $\rho $ of $\mathfrak {so}(0,3)$ or $\mathfrak {so}(1,2)$ :

(7.1) $$ \begin{align}\rho(A_1)=-\tfrac12\sigma_3,\quad \rho(A_2)=\tfrac12\sigma_2,\quad\rho(A_3)=-\tfrac12\varepsilon\sigma_1.\end{align} $$

Here, $(A_1,A_2,A_3)$ constitutes a basis for $\mathfrak {so}(0,3)$ when $\varepsilon =1$ and for $\mathfrak {so}(2,1)$ when $\varepsilon =-1$ . This can be extended to the representation of

$$ \begin{align*}\mathfrak{n}_{00}=\mathbb{R}\oplus \mathfrak{so}\big(\tfrac{1-\varepsilon}{2},\tfrac{5+\varepsilon}{2}\big)\end{align*} $$

in $S=\mathbb {R}^4$ by setting the value of $\rho $ on the generator $A_4=\mathrm {Id}$ as

(7.2) $$ \begin{align}\rho(A_4)=\tfrac12\mathfrak(I\otimes I).\end{align} $$

For this representation of $\mathbb {R}\oplus \mathfrak {so}\big (\tfrac {1-\varepsilon }{2},\tfrac {5+\varepsilon }{2}\big )$ , the magical equation (2.1) has a following solution:

$$ \begin{align*}\begin{aligned} \omega^1{}_{\mu\nu}=&\begin{pmatrix} 0&0&1&0\\0&0&0&-1\\-1&0&0&0\\0&1&0&0\end{pmatrix},\quad \omega^2{}_{\mu\nu}=\begin{pmatrix} 0&0&0&-1\\0&0&-1&0\\0&1&0&0\\1&0&0&0\end{pmatrix},\\ &\\ &\omega^1{}_{\mu\nu}=\begin{pmatrix} 0&-\varepsilon&0&0\\\varepsilon&0&0&0\\0&0&0&-1\\0&0&1&0\end{pmatrix}, \end{aligned}\end{align*} $$

with

(7.3) $$ \begin{align}\begin{aligned} \tau(A_1)=\begin{pmatrix} 0&0&0\\0&0&\varepsilon\\0&-1&0\end{pmatrix},\quad &\tau(A_2)=\begin{pmatrix} 0&0&-\varepsilon\\0&0&0\\-1&0&0\end{pmatrix},\quad\tau(A_3)=\begin{pmatrix} 0&-\varepsilon&0\\\varepsilon&0&0\\0&0&0\end{pmatrix}\\ &\tau(A_4)=\begin{pmatrix} 1&0&0\\0&1&0\\0&0&1\end{pmatrix}. \end{aligned} \end{align} $$

This in particular gives the vectorial representation $\tau $ of

$$ \begin{align*}\mathfrak{n}_{00}=\mathbb{R}\oplus \mathfrak{so}\big(\tfrac{1-\varepsilon}{2},\tfrac{5+\varepsilon}{2}\big)\end{align*} $$

in $R=\mathbb {R}^3$ .

Now, by using this solution for $(\rho ,\omega ,\tau )$ and applying our Corollary 4.1, we have an $(s=4)$ -dimensional manifold $N=\mathbb {R}^4$ , equipped with $r=3$ two-forms $\omega ^i=\tfrac 12\omega ^{}_{\mu \nu }\mathrm {d} x^\mu \wedge \mathrm {d} x^\nu $ , $i=1,2,3$ , which contactifies to an $(s+r)=7$ -dimensional manifold $M=\mathbb {R}^7$ having a distribution structure $(M,{\mathcal D})$ defined as an annihilator of the $r=3$ one-forms $\lambda ^i=\mathrm {d} u^i+\omega ^i{}_{\mu \nu }x^\mu \mathrm {d} x^\nu $ , $i=1,2,3$ . We have the following theorem.

Theorem 7.1. Let $M=\mathbb {R}^7$ with coordinates $(u^1,u^2,u^3,x^1,x^2,x^3,x^4)$ , and consider three 1-forms $\lambda ^1,\lambda ^2,\lambda ^3$ on M given by

$$ \begin{align*}\begin{aligned} \lambda^1=&\mathrm{d} u^1+x^1\mathrm{d} x^3-x^2\mathrm{d} x^4,\\ \lambda^2=&\mathrm{d} u^2-x^1\mathrm{d} x^4-x^2\mathrm{d} x^3,\\ \lambda^3=&\mathrm{d} u^3-\varepsilon x^1\mathrm{d} x^2-x^3\mathrm{d} x^4, \end{aligned}\quad\quad\mathrm{with}\quad\quad\varepsilon=\pm1.\end{align*} $$

The rank 4 distribution ${\mathcal D}$ on M defined as has its Lie algebra of infinitesimal authomorphism $\mathfrak {aut}(\mathcal D)$ isomorphic to the Tanaka prolongation of , where $(\rho ,S=\mathbb {R}^4)$ is the spinorial representation (7.1)-(7.2) of $\mathfrak {n}_{00}=\mathbb {R}\oplus \mathfrak {so}\big (\tfrac {1-\varepsilon }{2},\tfrac {5+\varepsilon }{2}\big )$ , and $(\tau ,R=\mathbb {R}^3)$ is the vectorial representation (7.3) of $\mathfrak {n}_{00}$ .

The symmetry algebra $\mathfrak {aut}({\mathcal D})$ is isomorphic to the simple Lie algebra $\mathfrak {sp}\big (\tfrac {1-\varepsilon }{2},\tfrac {5+\varepsilon }{2}\big )$ ,

$$ \begin{align*}\mathfrak{aut}({\mathcal D})=\mathfrak{sp}\big(\tfrac{1-\varepsilon}{2},\tfrac{5+\varepsilon}{2}\big),\end{align*} $$

having the following natural gradation:

$$ \begin{align*}\mathfrak{aut}({\mathcal D})=\mathfrak{n}_{-2}\oplus\mathfrak{n}_{-1}\oplus\mathfrak{n}_0\oplus\mathfrak{n}_1\oplus\mathfrak{n}_2,\end{align*} $$

with $\mathfrak {n}_{-2}=R$ , $\mathfrak {n}_{-1}=S$ ,

$$ \begin{align*}\begin{aligned} \mathfrak{n}_0&=\mathfrak{n}_{00}\oplus \mathfrak{so}\big(\tfrac{1-\varepsilon}{2},\tfrac{5+\varepsilon}{2}\big)=\\&\mathbb{R}\oplus \mathfrak{so}\big(\tfrac{1-\varepsilon}{2},\tfrac{5+\varepsilon}{2}\big)\oplus \mathfrak{so}\big(\tfrac{1-\varepsilon}{2},\tfrac{5+\varepsilon}{2}\big),\end{aligned}\end{align*} $$

$\mathfrak {n}_{1}=S^*$ , $\mathfrak {n}_{2}=R^*$ , which is inherited from the distribution structure $(M,{\mathcal D})$ . The duality signs $*$ at $R^*$ and $S^*$ above are with respect to the Killing form in $\mathfrak {sp}\big (\tfrac {1-\varepsilon }{2},\tfrac {5+\varepsilon }{2}\big )$ .

The contactification $(M,{\mathcal D})$ is locally a flat model for the parabolic geometry of type $\Big (\mathbf {Sp}\big (\tfrac {1-\varepsilon }{2},\tfrac {5+\varepsilon }{2}\big ),P\Big )$ related to the following crossed satake diagrams:

1. (1) in the case of $\varepsilon =1$ , and

2. (2) in the case of $\varepsilon =-1$ .

8. Application: Obtaining the exceptionals from contactifications of spin representations; the $\mathfrak {f}_4$ case

We will now explain the Cartan realisation of the simple exceptional Lie algebra $\mathfrak {f}_4$ in dimension $\mathbb {R}^{15}$ mentioned in the introduction.

The Satake diagrams for the real forms of the complex simple exceptional Lie algebra $\mathfrak {f}_4$ are as follows:

The first diagram corresponds to the compact real form of $\mathfrak {f}_4$ and is not interesting for us.Footnote ² The other two diagrams are interesting:

1. (1) The last, , corresponds to the split real form $\mathfrak {f}_I$ , and

2. (2) the middle one, , denoted by $\mathfrak {f}_{II}$ in [Reference Čap and Slovák11], is also interesting, since similarly to $\mathfrak {f}_I$ , it defines a parabolic geometry in dimension 15.

Crossing the last node on the right in the diagrams for $\mathfrak {f}_I$ or $\mathfrak {f}_{II}$ , as in or , we see that in both algebras there exist parabolic subalgebras $\mathfrak {p}_I$ or $\mathfrak {p}_{II}$ , respectively, of dimension 37, $\dim (\mathfrak {p}_I)=\dim (\mathfrak {p}_{II})=37$ . In both respective cases, these choices of parabolics define similar gradations in the corresponding real forms $\mathfrak {f}_I$ , $\mathfrak {f}_{II}$ , of the simple exceptional Lie $\mathfrak {f}_4$ :

$$ \begin{align*}\mathfrak{f}_A=\mathfrak{n}_{-2A}\oplus\mathfrak{n}_{-1A}\oplus\mathfrak{n}_{0A}\oplus\mathfrak{n}_{1A}\oplus\mathfrak{n}_{2A}\quad \mathrm{for}\quad A=I,II, \end{align*} $$

with

$$ \begin{align*}\mathfrak{n}_{-A}=\mathfrak{n}_{-2A}\oplus\mathfrak{n}_{-1A}\quad \mathrm{for}\quad A=I,II,\end{align*} $$

being 2-step nilpotent and having grading components $\mathfrak {n}_{-2A}$ and $\mathfrak {n}_{-1A}$ of respective dimension $r_A=7$ and $s_A=8$ ,

$$ \begin{align*}r_A=\dim(\mathfrak{n}_{-2A})=7,\quad\quad s_A=\dim(\mathfrak{n}_{-1A})=8\quad \mathrm{for}\quad A=I,II.\end{align*} $$

The Lie algebra $\mathfrak {n}_{0A}$ in the Tanaka prolongation of $\mathfrak {n}_{-A}$ up to $0^{th}$ order is

1. (1) $\mathfrak {n}_{0I}=\mathbb {R}\oplus \mathfrak {so}(4,3)$ in the case of $\mathfrak {f}_I$ , and

2. (2) $\mathfrak {n}_{0II}=\mathbb {R}\oplus \mathfrak {so}(0,7)$ in the case of $\mathfrak {f}_{II}$ .

Thus, from the analysis performed here, we see that there exist two different 2-step filtered structures $(M_I,{\mathcal D}_I)$ and $(M_{II},{\mathcal D}_{II})$ , both in dimension 15, with the respective $F_I$ -symmetric, or $F_{II}$ -symmetric flat models, realised on $M_I=F_I/P_I$ or $M_{II}=F_{II}/P_{II}$ . Here, $F_I$ and $F_{II}$ denote the real Lie groups whose Lie algebras are $\mathfrak {f}_I$ and $\mathfrak {f}_{II}$ , respectively. Similarly, $P_I$ and $P_{II}$ are parabolic subgroups of respective $F_I$ and $F_{II}$ , whose Lie algebras are $\mathfrak {p}_I$ and $\mathfrak {p}_{II}$ . Recalling that each of the real groups $\mathbf {SO}(4,3)$ and $\mathbf {SO}(0,7)$ has two real irreducible representations $\rho $ in dimension $s=8$ and $\tau $ in dimension $r=7$ , with the 8-dimensional representation $\rho $ being the spin representation of either $\mathbf {SO}(4,3)$ or $\mathbf {SO}(0,7)$ , we can now give the explicit realisations of the $\mathbf {F}_4$ -symmetric structures $(M_A,{\mathcal D}_A)$ for $A=I,II$ .

8.1. Cartan’s realisation of $\mathfrak {f}_I$

The plan is to start with the Lie algebra $\mathfrak {n}_{00}=\mathbb {R}\oplus \mathfrak {so}(4,3)$ , as in the crossed Satake diagram of $\mathfrak {f}_I$ , and its two representations:

• • a representaion $(\rho ,S=\mathbb {R}^8)$ , corresponding to the spin representation of $\mathbf {SO}(4,3)$ in $(s=8)$ -dimensional space $\mathfrak {n}_{-1}=S$ of real Pauli spinors, and

• • a representation $(\tau ,R=\mathbb {R}^7)$ , corresponding to the vectorial representation of $\mathbf {SO}(4,3)$ in $(r=7)$ -dimensional space $\mathfrak {n}_{-2}=R$ of vectors in $\mathbb {R}^{(4,3)}$ .

Having these two representations of $\mathfrak {n}_{00}=\mathbb {R}\oplus \mathfrak {so}(4,3)$ in the same basis, we will then solve the equations (2.1) for the map $\omega \in \mathrm {Hom}(\bigwedge ^2S,R)$ , which will give us the commutators between elements in $\mathfrak {n}_{-1}$ . This via Corollary 4.1 will provide the explicit realisation of the 15-dimensional contactification $(M,{\mathcal D})$ with the exception of simple Lie algebra $F_I$ as its symmetry.

Actually, the passage from $\rho $ to $\tau $ in the above plan is a bit tricky, since we need to have these representations expressed in the same basis. To handle this obstacle, we will start with the spin representation $\rho $ in the space of Pauli spinors S, and then we will use the fact that the skew representation $\rho \wedge \rho $ in the space of the bispinors $\bigwedge ^2S$ decomposes as

$$ \begin{align*}\textstyle \bigwedge^2S=\bigwedge_{21}\oplus\bigwedge_{7},\end{align*} $$

where $\bigwedge _{21}$ is the 21-dimensional adjoint representation of $\mathbf {SO}(4,3)$ and $\bigwedge _7$ is its 7-dimensional vectorial representation $\tau $ . In this way, we will have the two representations $(\rho ,S)$ and $(\tau ,R=\bigwedge _7)$ , expressed in the same basis $\{A_I\}$ of $\mathbb {R}\oplus \mathbf {SO}(4,3)$ , and we will apply the Corollary 4.1 to get the desired $F_I$ -symmetric contactification in dimension 15. On doing this, we will use notation from Section 5.

According to [Reference Trautman, Naber, Tsou and Françoise19], the real 8-dimensional representation of the Clifford algebra ${\mathcal C}\ell (4,3)$ is generated by the seven 8-dimensional Pauli matrices:

$$ \begin{align*}\begin{aligned} &\sigma_1=\sigma_x\otimes\sigma_x\otimes\sigma_x\\ &\sigma_2=\sigma_x\otimes\sigma_x\otimes\epsilon\\ &\sigma_3=\sigma_x\otimes\sigma_x\otimes\sigma_z\\ &\sigma_4=\sigma_x\otimes\epsilon\otimes I\\ &\sigma_5=\sigma_x\otimes\sigma_z\otimes I\\ &\sigma_6=\epsilon\otimes I\otimes I\\ &\sigma_7=\sigma_z\otimes I\otimes I. \end{aligned} \end{align*} $$

Using the identities (5.3), especially the one saying that $\epsilon ^2=-I$ , one easily finds that the seven Pauli matrices $\sigma _i$ , $i=1,2,\dots ,7$ , satisfy the Clifford algebra identity

$$ \begin{align*}\sigma_i\sigma_j+\sigma_j\sigma_i\,\,=\,\,2g_{ij}\,\,(I\otimes I \otimes I)\,,\quad\quad i,j=1,2,\dots,7,\end{align*} $$

with the coefficients $g_{ij}$ forming a diagonal $7\times 7$ matrix

$$ \begin{align*}\Big(\,\,g_{ij}\,\,\Big)\,\,=\,\,\mathrm{diag}\Big(1,-1,1,-1,1,-1,1\Big)\end{align*} $$

of signature $(4,3)$ . Thus, the 8-dimensional Pauli matrices $\sigma _i$ , $i=1,\dots ,7$ generate the Clifford algebra ${\mathcal C}\ell (4,3)$ , and in turn, by the general theory, as described in Section 5, they define the spin representation $\rho $ of $\mathfrak {so}(4,3)$ in an 8-dimensional real vector space $S=\mathbb {R}^8$ of Pauli(-Majorana) spinors.

8.1.1/ The spinorial representation of $\mathfrak {so}(4,3)$

To be more explicit, let $(i,j)$ be such that $1\leq i<j\leq 7$ , and let I be a function

(8.1) $$ \begin{align} I(i,j)=1+i+\tfrac12(j-3)j\end{align} $$

on such pairs. Note that the function I is a bijection between the 21 pairs $(i,j)$ and the set of 21 natural numbers $I=1,2,\dots , 21$ . Consider the twenty-one $8\times 8$ real matrices $\sigma _i\sigma _j$ with $1\leq i<j\leq 7$ , and a basis $\{A_I\}_{I=1}^{21}$ in the Lie algebra $\mathfrak {so}(4,3)$ . Then the spin representation $\rho $ of $\mathfrak {so}(4,3)$ is given by

$$ \begin{align*}\rho(A_{I(i,j)})=\tfrac12\sigma_i\sigma_j\quad\mathrm{with}\quad 1\leq i<j\leq7.\end{align*} $$

Explicitly, we have

(8.2) $$ \begin{align} \begin{array}{lll} \rho(A_1)=\tfrac12 I\otimes I \otimes\sigma_z,&\quad \rho(A_8)=\tfrac12 I\otimes \epsilon\otimes\epsilon,&\quad \rho(A_{15})=\tfrac12 \sigma_z\otimes \sigma_z \otimes I,\\ \rho(A_2)=\tfrac12 I\otimes I \otimes\epsilon, &\quad\rho(A_9)=\tfrac12 I\otimes \epsilon \otimes\sigma_z, &\quad \rho(A_{16})=\tfrac12 \epsilon\otimes \sigma_x \otimes\sigma_x, \\ \rho(A_3)=\tfrac12 I\otimes I \otimes\sigma_x, &\quad\rho(A_{10})=\tfrac12 I\otimes \sigma_x \otimes I,&\quad \rho(A_{17})=\tfrac12 \epsilon\otimes \sigma_x \otimes\epsilon,\\ \rho(A_4)=\tfrac12 I\otimes I \otimes\sigma_x,&\quad \rho(A_{11})=\tfrac12 \sigma_z\otimes \sigma_x \otimes\sigma_x,&\quad \rho(A_{18})=\tfrac12 \epsilon\otimes \sigma_x \otimes\sigma_z,\\ \rho(A_5)=\tfrac12 I\otimes \sigma_z \otimes\epsilon,&\quad \rho(A_{12})=\tfrac12 \sigma_z\otimes \sigma_x \otimes\epsilon,&\quad \rho(A_{19})=\tfrac12 \epsilon\otimes \epsilon \otimes I,\\ \rho(A_6)=\tfrac12 I\otimes \sigma_z \otimes\sigma_z,&\quad \rho(A_{13})=\tfrac12 \sigma_z\otimes \sigma_x\otimes\sigma_z,&\quad \rho(A_{20})=\tfrac12 \epsilon\otimes \sigma_z \otimes I,\\ \rho(A_7)=\tfrac12 I\otimes \epsilon \otimes\sigma_x, &\quad \rho(A_{14})=\tfrac12 \sigma_z\otimes \epsilon\otimes I, &\quad \rho(A_{21})=\tfrac12 \sigma_x\otimes I \otimes I. \end{array} \end{align} $$

The spin representation $\rho $ of $V_0=\mathbb {R}\oplus \mathfrak {so}(4,3)$ needs one generator more. Let us call it $\rho (A_{22})$ . We have

$$ \begin{align*}\rho(A_{22})=\tfrac12 I\otimes I\otimes I.\end{align*} $$

We determine the structure constants $c^K{}_{IJ}$ of $\mathbb {R}\oplus \mathfrak {so}(4,3)$ in the basis $A_I$ from

(8.3) $$ \begin{align} [\,\,\rho(A_I),\,\,\rho(A_J)\,\,]\,\,=\,\,c^K{}_{IJ}\,\,\rho(A_K).\end{align} $$

8.1.2. Obtaining the vectorial representation of $\mathfrak {so}(4,3)$

Now, we take the space $\bigwedge ^2S$ and consider the skew symmetric representation

$$ \begin{align*}{\stackrel{\scriptscriptstyle{a}}{\rho}}{}=\rho\wedge\rho\end{align*} $$

in it. We will write it in the standard basis $f_\mu $ , $\mu =1,\dots , 8$ in $S=\mathbb {R}^8$ . We have $\rho (A_I)f_\mu =\rho _I{}^\nu {}_\mu f_\nu $ . Now, the components of the 28-dimensional representation ${\stackrel {\scriptscriptstyle {a}}{\rho }}{}=\rho \wedge \rho _1$ are

$$ \begin{align*}{\stackrel{\scriptscriptstyle{a}}{\rho}}{}_{I}{}^{\mu\nu}{}_{\alpha\beta}\,\,=\,\,\,\,\rho_I{}^{\mu}{}_\alpha\delta^{\nu}{}_\beta+ \delta^{\mu}{}_\alpha\rho_I{}^{\nu}{}_\beta-\rho_I{}^{\nu}{}_\alpha\delta^{\mu}{}_\beta-\delta^{\nu}{}_\alpha\rho_I{}^{\mu}{}_\beta\,\,,\end{align*} $$

and we have

$$ \begin{align*}\big({\stackrel{\scriptscriptstyle{a}}{\rho}}{}(A_I)w\big){}^{\mu\nu}={\stackrel{\scriptscriptstyle{a}}{\rho}}{}_{I}{}^{\mu\nu}{}_{\alpha\beta}w^{\alpha\beta}, \quad \forall w^{\alpha\beta}=w^{[\alpha\beta]}.\end{align*} $$

The Casimir operator for this representation is

$$ \begin{align*}{\mathcal C}\,\,=\,\,10\,\,K^{IJ}\,\,{\stackrel{\scriptscriptstyle{a}}{\rho}}{}(A_I)\,{\stackrel{\scriptscriptstyle{a}}{\rho}}{}(A_J),\end{align*} $$

where $K^{IJ}$ is the inverse of the Killing form matrix $K_{IJ}=c^L{}_{IM}c^M{}_{JL}$ in the basis $A_I$ . Since for the Killing form to be nondegenerate we must restrict to the semisimple part of $\mathbb {R}\oplus \mathfrak {so}(4,3)$ , here the indices $I,J,K,L,M=1,2\dots ,21$ , and as always are summed over the repeated indices. One can check that in this basis of $\mathfrak {so}(4,3)$ , the Killing form matrix is diagonal and reads

$$ \begin{align*}\big(\,\, K_{IJ}\,\,\big)= 10\,\mathrm{diag}\Big(\,1,-1,1,1,-1,1,-1,1,-1,1,1,-1,1,-1,1,-1,1,-1,1,-1,1\,\Big). \end{align*} $$

The Casimir $\mathcal C$ defines the decomposition of the 28-dimensional reducible representation ${\stackrel {\scriptscriptstyle {a}}{\rho }}{}=\rho _1\wedge \rho _1$ onto

$$ \begin{align*}\textstyle \bigwedge^2S=\bigwedge_{21}\oplus\bigwedge_7,\end{align*} $$

where the 7-dimensional irreducible representation space $\bigwedge _7$ is the eigenspace of the Casimir operator consisting of eigen-bispinors with eigenvalue equal to 6,

$$ \begin{align*}\textstyle {\mathcal C}\,\,\big(\bigwedge_7\big)\,=6\,\bigwedge_7.\end{align*} $$

Explicitly, in the same basis $A_I$ , $I=1,2,\dots ,21$ as before, this 7-dimensional representation $(\tau ,R=\bigwedge _7)$ of the $\mathfrak {so}(4,3)$ Lie algebra is given by

(8.4) $$ \begin{align}\begin{aligned} &\tau(A_1)=E_{66}-E_{22},\\ &\tau(A_2)=\tfrac12(E_{23}-E_{32}+E_{25}-E_{52}+E_{36}-E_{63}+E_{56}-E_{65}),\\ &\tau(A_3)=\tfrac12(E_{23}+E_{32}+E_{25}+E_{52}+E_{36}+E_{63}+E_{56}+E_{65}),\\ &\tau(A_4)=\tfrac12(E_{23}+E_{32}-E_{25}-E_{52}-E_{36}-E_{63}+E_{56}+E_{65}),\\ &\tau(A_5)=\tfrac12(E_{23}-E_{32}-E_{25}+E_{52}-E_{36}+E_{63}+E_{56}-E_{65}),\\ &\tau(A_6)=E_{33}-E_{55},\\ &\tau(A_7)=\tfrac12(E_{12}-E_{21}-E_{16}+E_{61}-E_{27}+E_{72}+E_{67}-E_{76}),\\ &\tau(A_8)=\tfrac12(-E_{12}-E_{21}-E_{16}-E_{61}-E_{27}-E_{72}-E_{67}-E_{76}),\\ &\tau(A_9)=\tfrac12(E_{13}-E_{31}+E_{15}-E_{51}-E_{37}+E_{73}-E_{57}+E_{75}),\\ &\tau(A_{10})=\tfrac12(E_{13}+E_{31}-E_{15}-E_{51}+E_{37}+E_{73}-E_{57}-E_{75}),\\ &\tau(A_{11})=\tfrac12(-E_{12}-E_{21}+E_{16}+E_{61}+E_{27}+E_{72}-E_{67}-E_{76}),\\ &\tau(A_{12})=\tfrac12(E_{12}-E_{21}+E_{16}-E_{61}+E_{27}-E_{72}+E_{67}-E_{76}),\\ &\tau(A_{13})=\tfrac12(-E_{13}-E_{31}-E_{15}-E_{51}+E_{37}+E_{73}+E_{57}+E_{75}),\\ &\tau(A_{14})=\tfrac12(-E_{13}+E_{31}+E_{15}-E_{51}-E_{37}+E_{73}+E_{57}-E_{75}),\\ &\tau(A_{15})=E_{11}-E_{77},\\ &\tau(A_{16})=\tfrac12(2E_{24}-E_{42}+E_{46}-2E_{64}),\\ &\tau(A_{17})=\tfrac12(2E_{24}+E_{42}+E_{46}+2E_{64}),\\ &\tau(A_{18})=\tfrac12(2E_{34}-E_{43}-E_{45}+2E_{54}),\\ &\tau(A_{19})=\tfrac12(-2E_{34}-E_{43}+E_{45}+2E_{54}),\\ &\tau(A_{20})=\tfrac12(-2E_{14}+E_{41}+E_{47}-2E_{74}),\\ &\tau(A_{21})=\tfrac12(2E_{14}+E_{41}-E_{47}-2E_{74}),\\ &\tau(A_{22})=E_{11}+E_{22}+E_{33}+E_{44}+E_{55}+E_{66}+E_{77}, \end{aligned}\end{align} $$

where $E_{ij}$ , $i,j=1,2,\dots ,7$ denote $7\times 7$ matrices with zeroes everywhere except the value 1 in the entry $(i,j)$ seating at the crossing of the ith row and the jth column.

One can check that

$$ \begin{align*}[\,\,\tau(A_I),\,\,\tau(A_J)\,\,]\,\,=\,\,c^K{}_{IJ}\,\,\tau(A_K),\end{align*} $$

with the same structure constants as in (8.3).

8.1.3. A contactification with $\mathfrak {f}_I$ symmetry

So now we are in the situation of having two representations $(\rho ,S)$ and $(\tau ,R=\bigwedge _7)$ od $\mathfrak {n}_{00}=\mathbb {R}\oplus \mathfrak {so}(4,3)$ , and we can try to solve the equation (2.1) for the map $\omega \in \mathrm {Hom}(\bigwedge ^2S,R)$ . Of course, if we started with some arbitrary $\rho $ and $\tau $ , this equation would not have solutions other than 0, but here we expect to have solution since we know it from the Cartan’s PhD thesis [Reference Cartan8] and the announcement in Helgason’s paper [Reference Helgason13]. And indeed, there is a solution for a nonzero $\omega $ , which when written in the basis $\{f_\mu \}$ in S and $\{e_i\}$ in R is such that it gives the seven 2-forms $\omega ^i=\tfrac 12\omega ^i{}_{\mu \nu }\mathrm {d} x^\mu \wedge \mathrm {d} x^\nu $ , $i=1,\dots ,7$ , in $N=\mathbb {R}^8$ given by

(8.5) $$ \begin{align}\begin{aligned} \omega^1&=\mathrm{d} x^1\wedge\mathrm{d} x^2-\mathrm{d} x^7\wedge\mathrm{d} x^8,\\ \omega^2&=\mathrm{d} x^2\wedge\mathrm{d} x^4-\mathrm{d} x^6\wedge\mathrm{d} x^8,\\ \omega^3&=\mathrm{d} x^1\wedge\mathrm{d} x^4-\mathrm{d} x^5\wedge\mathrm{d} x^8,\\ \omega^4&=\tfrac12\,\big(\,\mathrm{d} x^1\wedge\mathrm{d} x^6-\mathrm{d} x^2\wedge\mathrm{d} x^5-\mathrm{d} x^3\wedge\mathrm{d} x^8+\mathrm{d} x^4\wedge\mathrm{d} x^7\,\big),\\ \omega^5&=\mathrm{d} x^2\wedge\mathrm{d} x^3-\mathrm{d} x^6\wedge\mathrm{d} x^7,\\ \omega^6&=\mathrm{d} x^1\wedge\mathrm{d} x^3-\mathrm{d} x^5\wedge\mathrm{d} x^7,\\ \omega^7&=\mathrm{d} x^3\wedge\mathrm{d} x^4-\mathrm{d} x^5\wedge\mathrm{d} x^6. \end{aligned}\end{align} $$

These, via the contactification and the theory summarised in Corollary 4.1, lead to the following theorem.

Theorem 8.1. Let $M=\mathbb {R}^{15}$ with coordinates $(u^1,\dots ,u^7,x^1,\dots ,x^8)$ , and consider seven 1-forms $\lambda ^i,\dots ,\lambda ^7$ on M given by

$$ \begin{align*}\begin{aligned} \lambda^1&=\mathrm{d} u^1+ x^1\mathrm{d} x^2- x^7\mathrm{d} x^8,\\ \lambda^2&=\mathrm{d} u^2+ x^2\mathrm{d} x^4- x^6\mathrm{d} x^8,\\ \lambda^3&=\mathrm{d} u^3+ x^1\mathrm{d} x^4- x^5\mathrm{d} x^8,\\ \lambda^4&=\mathrm{d} u^4+\tfrac12\,\big(\, x^1\mathrm{d} x^6- x^2\mathrm{d} x^5-x^3\mathrm{d} x^8+ x^4\mathrm{d} x^7\,\big),\\ \lambda^5&=\mathrm{d} u^5+x^2\mathrm{d} x^3- x^6\mathrm{d} x^7,\\ \lambda^6&=\mathrm{d} u^6+x^1\mathrm{d} x^3- x^5\mathrm{d} x^7,\\ \lambda^7&=\mathrm{d} u^7+x^3\mathrm{d} x^4- x^5\mathrm{d} x^6. \end{aligned}\end{align*} $$

The rank 8 distribution ${\mathcal D}$ on M defined as has its Lie algebra of infinitesimal authomorphism $\mathfrak {aut}(\mathcal D)$ isomorphic to the Tanaka prolongation of , where $(\rho ,S=\mathbb {R}^8)$ is the spinorial representation (8.2) of $\mathfrak {n}_{00}=\mathbb {R}\oplus \mathfrak {so}\big (4,3)$ and $(\tau ,R=\mathbb {R}^7)$ is the vectorial representation (8.4) of $\mathfrak {n}_{00}$ .

The symmetry algebra $\mathfrak {aut}({\mathcal D})$ is isomorphic to the simple Lie algebra $\mathfrak {f}_I$ ,

$$ \begin{align*}\mathfrak{aut}({\mathcal D})=\mathfrak{f}_I,\end{align*} $$

having the following natural gradation:

$$ \begin{align*}\mathfrak{aut}({\mathcal D})=\mathfrak{n}_{-2}\oplus\mathfrak{n}_{-1}\oplus\mathfrak{n}_0\oplus\mathfrak{n}_1\oplus\mathfrak{n}_2,\end{align*} $$

with $\mathfrak {n}_{-2}=R$ , $\mathfrak {n}_{-1}=S$ ,

$$ \begin{align*}\mathfrak{n}_0=\mathfrak{n}_{00}=\mathbb{R}\oplus\mathfrak{so}(4,3),\end{align*} $$

$\mathfrak {n}_{1}=S^*$ , $\mathfrak {n}_{2}=R^*$ , which is inherited from the distribution structure $(M,{\mathcal D})$ . The duality signs $*$ at $R^*$ and $S^*$ above are with respect to the Killing form in $\mathfrak {f}_I$ .

The contactification $(M,{\mathcal D})$ is locally a flat model for the parabolic geometry of type $(F_I,P_I)$ related to the following crossed Satake diagram of $\mathfrak {f}_I$ .

Remark 8.2. Please note that this is an example of an application of the magical equation (2.1) in which the starting algebra $\mathfrak {n}_{00}$ was big enough, so that its Tanaka prolongation $\mathfrak {n}_0$ counterpart is precisely equal to $\mathfrak {n}_{00}$ . This was actually expected from the construction based on the crossed Satake diagram, , which shows that $\mathfrak {n}_0$ of this parabolic geometry is precisely our $\mathfrak {n}_{00}=\mathbb {R}\oplus \mathfrak {so}(4,3)$ .

Remark 8.3. One sees that the distribution $\mathcal D$ in $\mathbb {R}^{15}$ with $\mathfrak {f}_I$ symmetry presented in Theorem 8.1 looks different that the distribution from our Example 1.3. It follows, however, that both these distributions are locally equivalent, and both have the same simple exceptional Lie algebra $\mathfrak {f}_I$ as an algebra of their authomorphisms.

8.1.4. Contactification for $\mathfrak {f}_I$ : more algebra about $\mathfrak {so}(4,3)$

In our construction of the $\mathfrak {f}_I$ symmetric distribution $\mathcal D$ in Theorem 8.1, the crucial role was played by the 7-dimensional span of 2-forms $\omega ^i$ , $i=1,2,\dots ,7$ . If we were given these seven 2-forms, we would produce the $\mathfrak {f}_I$ symmetric distribution $\mathcal D$ by the procedure of contactification.

It turns out that in $S=\mathbb {R}^8$ , there is a particular 4-formFootnote ³

$$ \begin{align*}\Phi=\tfrac{1}{4!}\Phi_{\mu\nu\rho\sigma}\mathrm{d} x^\mu\wedge\mathrm{d} x^\nu\wedge\mathrm{d} x^\rho\wedge \mathrm{d} x^\sigma\end{align*} $$

that is $\mathbb {R}\oplus \mathfrak {so}(4,3)$ invariant

$$ \begin{align*}\rho_I{}^\alpha{}_\mu \Phi_{\alpha\nu\rho\sigma}+\rho_I{}^\alpha{}_\nu \Phi_{\mu\alpha\rho\sigma}+\rho_I{}^\alpha{}_\rho \Phi_{\mu\nu\alpha\sigma}+\rho_I{}^\alpha{}_\sigma \Phi_{\mu\nu\rho\alpha}=S_I\Phi_{\mu\nu\rho\sigma}.\end{align*} $$

It may be represented by

$$ \begin{align*}\Phi=h_{ij}\omega^i\wedge\omega^j,\end{align*} $$

where $\omega ^i$ are given by (8.5) and

$$ \begin{align*}\big(\,\,h_{ij}\,\,\big)\,\,=\,\,\tfrac12\,\, \begin{pmatrix} 0&0&0&0&0&0&1\\0&0&0&0&0&-1&0\\0&0&0&0&1&0&0\\0&0&0&1&0&0&0\\0&0&1&0&0&0&0\\0&-1&0&0&0&0&0\\1&0&0&0&0&0&0 \end{pmatrix},\end{align*} $$

or in wordsFootnote ⁴ : $h_{ij}$ , $i,j,=1,2,\dots ,7$ , are all zero except $h_{17}=h_{71}=-h_{26}=-h_{62}=h_{35}=h_{53}=h_{44}=1$ .

The form $\Phi $ in full beauty reads

(8.6) $$ \begin{align}\begin{aligned} \tfrac23\Phi\,\,=\,\,2\,\,\Big(\,\,&\mathrm{d} x^1\wedge\mathrm{d} x^2\wedge\mathrm{d} x^3\wedge\mathrm{d} x^4+\mathrm{d} x^5\wedge\mathrm{d} x^6\wedge\mathrm{d} x^7\wedge\mathrm{d} x^8\,\,\Big)-\\ &\mathrm{d} x^1\wedge\mathrm{d} x^2\wedge\mathrm{d} x^5\wedge\mathrm{d} x^6+\mathrm{d} x^1\wedge\mathrm{d} x^3\wedge\mathrm{d} x^6\wedge\mathrm{d} x^8-\\&\mathrm{d} x^1\wedge\mathrm{d} x^4\wedge\mathrm{d} x^6\wedge\mathrm{d} x^7-\mathrm{d} x^2\wedge\mathrm{d} x^3\wedge\mathrm{d} x^5\wedge\mathrm{d} x^8+\\&\mathrm{d} x^2\wedge\mathrm{d} x^4\wedge\mathrm{d} x^5\wedge\mathrm{d} x^7-\mathrm{d} x^3\wedge\mathrm{d} x^4\wedge\mathrm{ d} x^7\wedge\mathrm{d} x^8. \end{aligned} \end{align} $$

Remark 8.4. It is remarkable that this 4-form alone encaptures all the features of the $\mathfrak {f}_I$ symmetric contactification we discussed in the entire Section 8.1. By this, we mean following:

1. (1) Consider $N=\mathbb {R}^8$ with coordinates $(x^\mu )$ , $\mu =1,2,\dots ,8$ and the 4-form

   $$ \begin{align*}\Phi=\tfrac{1}{4!}\Phi_{\mu\nu\rho\sigma}\mathrm{d} x^\mu\wedge\mathrm{d} x^\nu\wedge\mathrm{d} x^\rho\wedge \mathrm{ d} x^\sigma\end{align*} $$
   given by (8.6).

2. (2) Consider an equation

   $$ \begin{align*}A^\alpha{}_\mu \Phi_{\alpha\nu\rho\sigma}+A^\alpha{}_\nu \Phi_{\mu\alpha\rho\sigma}+A^\alpha{}_\rho \Phi_{\mu\nu\alpha\sigma}+A^\alpha{}_\sigma \Phi_{\mu\nu\rho\alpha}=S\Phi_{\mu\nu\rho\sigma}\end{align*} $$
   for the real $7\times 7$ matrix $A=(A^\mu {}_\nu )$ .

3. (3) For simplicity, solve it in two steps:

   • • First with $S=0$ . You obtain 21-dimensional solution space, which will be the spin representation $\rho $ of $\mathfrak {so}(4,3)$ . It is given $\rho (A)=A$ .

   • • Then prove that the only solution with $S\neq 0$ corresponds to $S=4$ , and that, modulo the addition of linear combinations of solutions with $S=0$ , it is given by $A=\mathrm {Id}_{8\times 8}$ . Extend your possible As with $S=0$ to As including $A=\mathrm {Id}_{8\times 8}$ .

4. (4) In this way, you will show that the stabiliser in $\mathfrak {gl}(8,\mathbb {R})$ of the 4-form $\Phi $ is the Lie algebra $\mathbb {R}\oplus \mathfrak {so}(4,3)$ in the spin representation $\rho $ of Pauli spinors; $\rho (A)=A$ .

5. (5) Then search for a 7-dimensional space of 2-forms spanned say by the 2-forms $\omega ^i=\tfrac 12\omega ^i{}_{\mu \nu }\mathrm {d} x^\mu \wedge \mathrm {d} x^\nu $ satisfying

   $$ \begin{align*}A^\alpha{}_\mu\,\, \omega^i{}_{\alpha\nu}\,\,+\,\,A^\alpha{}_\nu\,\, \omega^i{}_{\mu\alpha}\,\,=\,\,s^i{}_j \,\,\omega^j{}_{\mu\nu}\,\,\end{align*} $$
   for all As from the spin representation $\rho (A)=A$ of $\mathbb {R}\oplus \mathfrak {so}(4,3)$ . Here, $s^i{}_j$ are auxiliary constantsFootnote ⁵ .

6. (6) This space is uniquely defined by these equations, and after solving them, you will get 7 linearly independent 2-forms $(\omega ^1,\dots ,\omega ^7)$ in $N=\mathbb {R}^8$ .

7. (7) Contactifying the resulting structure $\big (N,\mathrm {Span}(\omega ^1,\dots ,\omega ^7)\big )$ , as we did, for example, in Theorem 8.1, you will get $\mathfrak {f}_I$ symmetric distribution ${\mathcal D}$ in $\mathbb {R}^7\to \big (\,M=\mathbb {R}^{15}\,\big )\to \big (\,N=\mathbb {R}^8\,\big )$ .

8.2. Realisation of $\mathfrak {f}_{II}$

It seems that Cartan was only interested in the explicit realisation of $\mathfrak {f}_I$ . The realisation of $\mathfrak {f}_{II}$ can be obtained in the same spirit as we have described in Section 8.1. Here, without much of the explanations since they parallel Section 8.1, we only display the main steps leading to this realisation.

We start with the representation of the Clifford algebra ${\mathcal C}\ell (0,7)$ generated by the seven $\rho $ -matrices from (5.4). They satisfy

$$ \begin{align*}\rho_i\rho_j+\rho_j\rho_i=-2\delta_{ij}I_{8\times 8},\quad i,j=1,\dots,7.\end{align*} $$

They induce the 8-dimensional representation

$$ \begin{align*}\rho:\mathbb{R}\oplus\mathfrak{so}(0,7)\to \mathrm{End}(S)\end{align*} $$

of $n_{00}=\mathbb {R}\oplus \mathfrak {so}(0,7)$ in the space $S=\mathbb {R}^8$ of real Pauli spinors, generated by the 22 real $8\times 8$ matrices:

$$ \begin{align*}\begin{aligned} &\rho(A_{I(i,j)})=\tfrac12\rho_i\rho_j, \quad 1\leq i<j\leq 7,\\ &\rho(A_{22})=\tfrac12 (I\otimes I\otimes I), \end{aligned} \end{align*} $$

with the index $I=I(i,j)$ given by (8.1) and with $I,\sigma _x,\epsilon ,\sigma _z$ given by (5.1)–(5.2). Explicitly, in terms of matrices $I,\sigma _x,\epsilon ,\sigma _z$ , the generators of this spinorial representation of $\mathfrak {so}(0,7)$ are

(8.7) $$ \begin{align} \begin{array}{lll} \rho(A_1)=-\tfrac12 I\otimes \epsilon \otimes\sigma_z,&\quad \rho(A_8)=\tfrac12 \epsilon\otimes \sigma_z\otimes\sigma_z,&\quad \rho(A_{15})=-\tfrac12 I\otimes \epsilon \otimes I,\\ \rho(A_2)=\tfrac12 I\otimes \epsilon \otimes\sigma_x, &\quad\rho(A_9)=-\tfrac12 \epsilon\otimes \sigma_z \otimes\sigma_x, &\quad \rho(A_{16})=-\tfrac12 \sigma_x\otimes I \otimes\epsilon, \\ \rho(A_3)=-\tfrac12 I\otimes I \otimes\epsilon, &\quad\rho(A_{10})=-\tfrac12 I\otimes \sigma_z \otimes \epsilon,&\quad \rho(A_{17})=-\tfrac12 \sigma_x\otimes\epsilon \otimes\sigma_x,\\ \rho(A_4)=-\tfrac12 \epsilon\otimes \epsilon\otimes\epsilon,&\quad \rho(A_{11})=\tfrac12 \epsilon\otimes \sigma_z \otimes I,&\quad \rho(A_{18})=-\tfrac12 \sigma_x\otimes \epsilon \otimes\sigma_z,\\ \rho(A_5)=\tfrac12 \epsilon\otimes I \otimes\sigma_x,&\quad \rho(A_{12})=-\tfrac12 \epsilon\otimes \sigma_x \otimes\sigma_z,&\quad \rho(A_{19})=\tfrac12 \sigma_z\otimes \epsilon \otimes I,\\ \rho(A_6)=\tfrac12 \epsilon\otimes I\otimes\sigma_z,&\quad \rho(A_{13})=\tfrac12 \epsilon\otimes \sigma_x\otimes\sigma_x,&\quad \rho(A_{20})=\tfrac12 \sigma_z\otimes \sigma_x \otimes \epsilon,\\ \rho(A_7)=\tfrac12 \epsilon\otimes \sigma_x \otimes I, &\quad \rho(A_{14})=\tfrac12 I \otimes \sigma_x\otimes \epsilon, &\quad \rho(A_{21})=\tfrac12 \sigma_z\otimes \sigma_z \otimes \epsilon. \end{array} \end{align} $$

We also write down the corresponding generators of the vectorial representation $\tau $ , which is the 7-dimensional irreducible component $\bigwedge _7$ of the representation $\rho \wedge \rho $ , which decomposes as $\bigwedge ^2S=\bigwedge _{21}\oplus \bigwedge _7$ . These generators read

(8.8) $$ \begin{align} \begin{array}{lll} \tau(A_1)=E_{31}-E_{13},&\quad \tau(A_8)=E_{37}-E_{73},&\quad \tau(A_{15})=E_{75}-E_{57},\\ \tau(A_2)=E_{12}-E_{21}, &\quad\tau(A_9)=E_{72}-E_{27}, &\quad \tau(A_{16})=E_{14}-E_{41}, \\ \tau(A_3)=E_{32}-E_{23}, &\quad\tau(A_{10})=E_{76}-E_{67},&\quad \tau(A_{17})=E_{34}-E_{43},\\ \tau(A_4)=E_{61}-E_{16},&\quad \tau(A_{11})=E_{51}-E_{15},&\quad \tau(A_{18})=E_{42}-E_{24},\\ \tau(A_5)=E_{36}-E_{63},&\quad \tau(A_{12})=E_{53}-E_{35},&\quad \tau(A_{19})=E_{46}-E_{64},\\ \tau(A_6)=E_{62}-E_{26},&\quad \tau(A_{13})=E_{25}-E_{52},&\quad \tau(A_{20})=E_{47}-E_{74},\\ \tau(A_7)=E_{17}-E_{71}, &\quad \tau(A_{14})= E_{65}-E_{56},&\quad \tau(A_{21})=E_{54}-E_{45}, \end{array} \end{align} $$

where $E_{ij}$ are $7\times 7$ matrices with all zero entries, except at the ith-jth entry, where 1 resides.

We are again in a position ready for application of our Lemma 2.1. Given the representations $(\rho ,S=\mathbb {R}^8)$ and $(\tau ,R=\bigwedge _7)$ of $\mathfrak {so}(0,7)$ , we solve the magical equation (2.1) for $\omega =\tfrac 12\omega ^i{}_{\mu \nu }e_i\otimes f^\mu \wedge f^\nu $ . In this way, we obtain the seven 2-forms $\omega ^i=-\tfrac 12\omega ^i{}_{\mu \nu }\mathrm {d} x^\mu \wedge \mathrm {d} x^\nu $ on $N=\mathbb {R}^8$ , with coordinates $(x^\mu )_{\mu =1}^8$ , which read as follows:

(8.9) $$ \begin{align}\begin{aligned} \omega^1&=-\mathrm{d} x^1\wedge\mathrm{d} x^2-\mathrm{d} x^3\wedge\mathrm{d} x^4+\mathrm{d} x^5\wedge\mathrm{d} x^6+\mathrm{d} x^7\wedge\mathrm{d} x^8,\\ \omega^2&=\mathrm{d} x^1\wedge\mathrm{d} x^3-\mathrm{d} x^2\wedge\mathrm{d} x^4-\mathrm{d} x^5\wedge\mathrm{d} x^7+\mathrm{d} x^6\wedge\mathrm{d} x^8,\\ \omega^3&=-\mathrm{d} x^1\wedge\mathrm{d} x^4-\mathrm{d} x^2\wedge\mathrm{d} x^3+\mathrm{d} x^5\wedge\mathrm{d} x^8+\mathrm{d} x^6\wedge\mathrm{d} x^7,\\ \omega^4&=\mathrm{d} x^1\wedge\mathrm{d} x^5+\mathrm{d} x^2\wedge\mathrm{d} x^6+\mathrm{d} x^3\wedge\mathrm{d} x^7+\mathrm{d} x^4\wedge\mathrm{d} x^8,\\ \omega^5&=-\mathrm{d} x^1\wedge\mathrm{d} x^6+\mathrm{d} x^2\wedge\mathrm{d} x^5+\mathrm{d} x^3\wedge\mathrm{d} x^8-\mathrm{d} x^4\wedge\mathrm{d} x^7,\\ \omega^6&=\mathrm{d} x^1\wedge\mathrm{d} x^7+\mathrm{d} x^2\wedge\mathrm{d} x^8-\mathrm{d} x^3\wedge\mathrm{d} x^5-\mathrm{d} x^4\wedge\mathrm{d} x^6,\\ \omega^7&=\mathrm{d} x^1\wedge\mathrm{d} x^8-\mathrm{d} x^2\wedge\mathrm{d} x^7+\mathrm{d} x^3\wedge\mathrm{d} x^6-\mathrm{d} x^4\wedge\mathrm{d} x^5. \end{aligned}\end{align} $$

These, via the contactification, lead to the following theorem.

Theorem 8.5. Let $M=\mathbb {R}^{15}$ with coordinates $(u^1,\dots ,u^7,x^1,\dots ,x^8)$ , and consider seven 1-forms $\lambda ^i,\dots ,\lambda ^7$ on M given by

$$ \begin{align*}\begin{aligned} \lambda^1&=\mathrm{d} u^1- x^1\mathrm{d} x^2- x^3\mathrm{d} x^4+ x^5\mathrm{d} x^6+ x^7\mathrm{d} x^8,\\ \lambda^2&=\mathrm{d} u^2+ x^1\mathrm{d} x^3- x^2\mathrm{d} x^4-x^5\mathrm{d} x^7+x^6\mathrm{d} x^8,\\ \lambda^3&=\mathrm{d} u^3- x^1\mathrm{d} x^4- x^2\mathrm{d} x^3+ x^5\mathrm{d} x^8+ x^6\mathrm{d} x^7,\\ \lambda^4&=\mathrm{d} u^4+ x^1\mathrm{d} x^5+ x^2\mathrm{d} x^6+ x^3\mathrm{d} x^7+ x^4\mathrm{d} x^8,\\ \lambda^5&=\mathrm{d} u^5- x^1\mathrm{d} x^6+x^2\mathrm{d} x^5+x^3\mathrm{d} x^8-x^4\mathrm{d} x^7,\\ \lambda^6&=\mathrm{d} u^6+ x^1\mathrm{d} x^7+ x^2\mathrm{d} x^8- x^3\mathrm{d} x^5-x^4\mathrm{d} x^6,\\ \lambda^7&=\mathrm{d} u^7+ x^1\mathrm{d} x^8- x^2\mathrm{d} x^7+ x^3\mathrm{d} x^6 - x^4\mathrm{d} x^5. \end{aligned}\end{align*} $$

The rank 8 distribution ${\mathcal D}$ on M defined as has its Lie algebra of infinitesimal authomorphisms $\mathfrak {aut}(\mathcal D)$ isomorphic to the Tanaka prolongation of , where $(\rho ,S=\mathbb {R}^8)$ is the spinorial representation (8.7) of $\mathfrak {n}_{00}=\mathbb {R}\oplus \mathfrak {so}\big (0,7)$ , and $(\tau ,R=\mathbb {R}^7)$ is the vectorial representation (8.8) of $\mathfrak {n}_{00}$ .

The symmetry algebra $\mathfrak {aut}({\mathcal D})$ is isomorphic to the simple Lie algebra $\mathfrak {f}_{II}$ ,

$$ \begin{align*}\mathfrak{aut}({\mathcal D})=\mathfrak{f}_{II},\end{align*} $$

having the following natural gradation

$$ \begin{align*}\mathfrak{aut}({\mathcal D})=\mathfrak{n}_{-2}\oplus\mathfrak{n}_{-1}\oplus\mathfrak{n}_0\oplus\mathfrak{n}_1\oplus\mathfrak{n}_2,\end{align*} $$

with $\mathfrak {n}_{-2}=R$ , $\mathfrak {n}_{-1}=S$ ,

$$ \begin{align*}\mathfrak{n}_0=\mathfrak{n}_{00}=\mathbb{R}\oplus\mathfrak{so}(0,7),\end{align*} $$

$\mathfrak {n}_{1}=S^*$ , $\mathfrak {n}_{2}=R^*$ , which is inherited from the distribution structure $(M,{\mathcal D})$ . The duality signs $*$ at $R^*$ and $S^*$ above are with respect to the Killing form in $\mathfrak {f}_{II}$ .

The contactification $(M,{\mathcal D})$ is locally a flat model for the parabolic geometry of type $(F_{II},P_{II})$ related to the following crossed Satake diagram of $\mathfrak {f}_{II}$ .

Remark 8.6. In this way, we realised the real form $\mathfrak {f}_{II}$ of the simple exceptional complex Lie algebra $\mathfrak {f}_4$ in $M=\mathbb {R}^{15}$ as a symmetry algebra of the Pfaffian system $(\lambda ^1,\dots ,\lambda ^7)$ . This realisation does not appear in Cartan’s theses.

Remark 8.7. Our present case of $\mathfrak {f}_{II}$ also admits description in terms of a certain $\mathbb {R}\oplus \mathfrak {so}(0,7)$ invariant 4-form $\Phi $ in $S=\mathbb {R}^8$ , analogous to the 4-form $\Phi $ introduced in Section 8.1.4, when we discussed $\mathfrak {f}_I$ . Skipping the details, we only mention that now $\Phi $ may be represented by

$$ \begin{align*}\Phi=h_{ij}\omega^i\wedge\omega^j,\end{align*} $$

where $\omega ^i$ are given by (8.9) and

$$ \begin{align*}\big(\,\,h_{ij}\,\,\big)= \begin{pmatrix} -1&0&0&0&0&0&0\\0&-1&0&0&0&0&0\\0&0&-1&0&0&0&0\\0&0&0&-1&0&0&0\\0&0&0&0&-1&0&0\\0&0&0&0&0&-1&0\\0&0&0&0&0&0&-1 \end{pmatrix}.\end{align*} $$

Explicitly, the form $\Phi $ reads

(8.10) $$ \begin{align}\begin{aligned} -\tfrac16\Phi\,\,=\,\,&\mathrm{d} x^1\wedge\mathrm{d} x^2\wedge\mathrm{d} x^3\wedge\mathrm{d} x^4-\mathrm{d} x^1\wedge\mathrm{d} x^2\wedge\mathrm{d} x^5\wedge\mathrm{d} x^6-\\ &\mathrm{d} x^1\wedge\mathrm{d} x^2\wedge\mathrm{d} x^7\wedge\mathrm{d} x^8-\mathrm{d} x^1\wedge\mathrm{d} x^3\wedge\mathrm{d} x^5\wedge\mathrm{d} x^7+\\&\mathrm{d} x^1\wedge\mathrm{d} x^3\wedge\mathrm{d} x^6\wedge\mathrm{d} x^8-\mathrm{d} x^1\wedge\mathrm{d} x^4\wedge\mathrm{d} x^5\wedge\mathrm{d} x^8-\\&\mathrm{d} x^1\wedge\mathrm{d} x^4\wedge\mathrm{d} x^6\wedge\mathrm{d} x^7-\mathrm{d} x^2\wedge\mathrm{d} x^3\wedge\mathrm{ d} x^5\wedge\mathrm{d} x^8-\\ &\mathrm{d} x^2\wedge\mathrm{d} x^3\wedge\mathrm{d} x^6\wedge\mathrm{d} x^7+\mathrm{d} x^2\wedge\mathrm{d} x^4\wedge\mathrm{d} x^5\wedge\mathrm{d} x^7-\\&\mathrm{d} x^2\wedge\mathrm{d} x^4\wedge\mathrm{d} x^6\wedge\mathrm{d} x^8-\mathrm{d} x^3\wedge\mathrm{d} x^4\wedge\mathrm{d} x^5\wedge\mathrm{d} x^6-\\&\mathrm{d} x^3\wedge\mathrm{d} x^4\wedge\mathrm{d} x^7\wedge\mathrm{d} x^8+\mathrm{d} x^5\wedge\mathrm{d} x^6\wedge\mathrm{ d} x^7\wedge\mathrm{d} x^8. \end{aligned} \end{align} $$

This 4-form alone encaptures all the features of the $\mathfrak {f}_{II}$ symmetric contactification we discussed in the entire Section 8.2. In particular, analogous statements as in Remark 8.4, with now $\mathfrak {so}(4,3)$ replaced by $\mathfrak {so}(0,7)$ , apply to the present 4-form $\Phi $ .

9. Spinorial representations in dimension 8

Dimension eight is quite exceptional, as, for example, 8 is the highest possible dimension for the existence of Euclidean Hurwitz algebras, gifting us with the algebra of octonions. From the perspective of our paper, which meanders through the realm of simple Lie algebras, eight is very special: among all the complex simple Lie algebras, the Dynkin diagram of $\mathfrak {d}_4=\mathfrak {so}(8,\mathbb {C})$ which is defined in dimension eight, is the most symmetric:

Visibly it has a threefold symmetry $S_3$ .

The Lie algebra $\mathfrak {so}(8,\mathbb {C})$ has six real forms. These are: $\mathfrak {so}(8,0)$ , $\mathfrak {so}(7,1)$ , $\mathfrak {so}(6,2)$ , $\mathfrak {so}^*(8)$ , $\mathfrak {so}(5,3)$ and $\mathfrak {so}(4,4)$ , with the following respective Satake diagrams:

We see that among these Satake diagrams, the only ones that share the $S_3$ symmetry of the Dynkin diagram of the complex algebra $\mathfrak {d}_4$ are those of the compact real form $\mathfrak {so}(8,0)$ and of the split real form $\mathfrak {so}(4,4)$ .

This $S_3$ symmetry of these two diagrams indicates that the lowest dimensional real representations of $\mathfrak {so}(4,4)$ and $\mathfrak {so}(8,0)$ may have additional features when compared with spinorial representations of other $\mathfrak {so}(p,q)$ s. In particular, for both $\mathfrak {so}(4,4)$ and $\mathfrak {so}(8,0)$ , we have the following:

• • Their Dirac representation $(\rho ,S)$ in the real vector space $S=\mathbb {R}^{16}$ is reducible over $\mathbb {R}$ and is split into two real Weyl representations $(\rho _+,S_+)$ and $(\rho _-,S_-)$ in the respective vector spaces of Weyl spinors $S_+=\mathbb {R}^8$ and $S_-=\mathbb {R}^8$ , which have the same real dimension eight,

  $$ \begin{align*}\rho=\rho_+\oplus\rho_-\quad\mathrm{in}\quad S=S_+\oplus S_-, \quad \mathrm{dim}_{\mathbb{R}} S_\pm=8.\end{align*} $$

• • The real Weyl representations $(\rho _\pm ,S_\pm )$ are faithful, irreducible and nonequivalent.

• • The defining representations $(\tau ,R)$ of $\mathfrak {so}(4,4)$ and $\mathfrak {so}(8,0)$ , as the algebra of endomorphisms in the space $R=\mathbb {R}^8$ of vectors preserving the bilinear form of respective signatures $(4,4)$ and $(8,0)$ has the same dimension eight as the two Weyl representations $(\rho _\pm ,S_\pm )$ .

• • The real defining representations $(\tau ,R)$ are irreducible for both $\mathfrak {so}(4,4)$ and $\mathfrak {so}(8,0)$ .

• • All three real 8-dimensional irreducible representations $(\rho _+,S_+)$ , $(\rho _-,S_-)$ and $(\tau ,R)$ of, respectively, both $\mathfrak {so}(4,4)$ and $\mathfrak {so}(8,0)$ are pairwise nonequivalent.

Thus, the Lie algebras $\mathfrak {so}(4,4)$ and $\mathfrak {so}(8,0)$ have three real, irreducible and nonequivalent representations $(\rho _+,\rho _-,\tau )$ in the vector space $\mathbb {R}^8$ of the defining dimension $p+q=8$ . For all $\mathfrak {so}(p,q)$ Lie algebras, this is the only dimension $p+q$ that such situation with the irreducible representations occurs.

Below, we provide the explicit description of the triality representations $(\rho _+,\rho _-,\tau )$ separately for $\mathfrak {so}(4,4)$ and $\mathfrak {so}(8,0)$ .

9.1. Triality representations of $\mathfrak {so}(4,4)$

We recall from Section 5 that the Lie algebra $\mathfrak {so}(4,4)$ admits a representation $\rho $ in the 16-dimensional real vector space $S=\mathbb {R}^{16}$ of Dirac spinors. This is obtained by using the Dirac $\gamma $ matrices generating the representation of the Clifford algebra ${\mathcal C}\ell (4,4)$ . In terms of the 2-dimensional Pauli matrices $(\sigma _x,\epsilon ,\sigma _z,I)$ , these look as follows:

(9.1) $$ \begin{align} \begin{aligned} &\gamma_1=\sigma_x\otimes\sigma_x\otimes\sigma_x\otimes\sigma_x\\ &\gamma_2=\sigma_x\otimes\sigma_x\otimes\sigma_x\otimes\epsilon\\ &\gamma_3=\sigma_x\otimes\sigma_x\otimes\sigma_x\otimes\sigma_z\\ &\gamma_4=\sigma_x\otimes\sigma_x\otimes\epsilon\otimes I\\ &\gamma_5=\sigma_x\otimes\sigma_x\otimes\sigma_z\otimes I\\ &\gamma_6=\sigma_x\otimes\epsilon\otimes I\otimes I\\ &\gamma_7=\sigma_x\otimes\sigma_z\otimes I\otimes I\\ &\gamma_8=\epsilon\otimes I\otimes I\otimes I. \end{aligned}\end{align} $$

They satisfy the Dirac identity

(9.2) $$ \begin{align} \gamma_i\gamma_j+\gamma_j\gamma_i=2g_{ij} (I\otimes I\otimes I\otimes I), \quad i,j=1,\dots,8,\end{align} $$

with

$$ \begin{align*}\big( g_{ij} \big)=\mathrm{diag}(1,-1,1,-1,1,-1,1,-1).\end{align*} $$

The 28 generators of $\mathfrak {so}(4,4)$ in the Majorana-Dirac spinor representation $\rho $ in the space of Dirac spinors $S=\mathbb {R}^{16}$ are given by

$$ \begin{align*}\rho(A_{I(i,j)})=\tfrac12\gamma_i\gamma_j, \quad 1\leq i<j\leq 8,\end{align*} $$

where we again have used the function $I=I(i,j)$ defined in (8.1). Note that since now $i<j$ can run from 1 to 8, the function has a range from 1 to 28. We add to these generators the scaling generator, $\rho (A_{29})$ ,

$$ \begin{align*}\rho(A_{29})=\tfrac12 I\otimes I\otimes I\otimes I.\end{align*} $$

This extends the Dirac representation $\rho $ of the Lie algebra $\mathfrak {so}(4,4)$ to the representation of the homothety Lie algebra $\mathfrak {co}(4,4)=\mathbb {R}\oplus \mathfrak {so}(4,4)$ .

In terms of the 2-dimensional Pauli matrices, these generators look like

(9.3) $$ \begin{align} \begin{array}{ll} \rho(A_1)= \tfrac12 I \otimes I\otimes I\otimes \sigma_z,&\rho(A_{15})=\tfrac12 I \otimes \sigma_z\otimes \sigma_z\otimes I,\\ \rho(A_2)= \tfrac12 I \otimes I\otimes I\otimes \epsilon ,&\rho(A_{16})=\tfrac12 I \otimes \epsilon\otimes \sigma_x\otimes \sigma_x,\\ \rho(A_3)= \tfrac12 I \otimes I\otimes I\otimes \sigma_x ,&\rho(A_{17})=\tfrac12 I \otimes \epsilon\otimes \sigma_x\otimes \epsilon ,\\ \rho(A_4)=\tfrac12 I \otimes I\otimes \sigma_z\otimes \sigma_x,&\rho(A_{18})=\tfrac12 I \otimes \epsilon\otimes \sigma_x\otimes \sigma_z ,\\ \rho(A_5)= \tfrac12 I \otimes I\otimes \sigma_z\otimes \epsilon,&\rho(A_{19})=\tfrac12 I \otimes \epsilon\otimes \epsilon\otimes I ,\\ \rho(A_6)= \tfrac12 I \otimes I\otimes \sigma_z\otimes \sigma_z,&\rho(A_{20})=\tfrac12 I \otimes \epsilon\otimes \sigma_z\otimes I ,\\ \rho(A_7)= \tfrac12 I \otimes I\otimes \epsilon\otimes \sigma_x,&\rho(A_{21})=\tfrac12 I \otimes \sigma_x\otimes I\otimes I,\\ \rho(A_8)= \tfrac12 I \otimes I\otimes \epsilon\otimes \epsilon,&\rho(A_{22})=\tfrac12 \sigma_z \otimes \sigma_x\otimes \sigma_x\otimes \sigma_x ,\\ \rho(A_9)= \tfrac12 I \otimes I\otimes \epsilon\otimes \sigma_z,&\rho(A_{23})=\tfrac12 \sigma_z \otimes \sigma_x\otimes \sigma_x\otimes \epsilon ,\\ \rho(A_{10})=\tfrac12 I \otimes I\otimes \sigma_x\otimes I,&\rho(A_{24})=\tfrac12 \sigma_z \otimes \sigma_x\otimes \sigma_x\otimes \sigma_z ,\\ \rho(A_{11})= \tfrac12 I \otimes \sigma_z\otimes \sigma_x\otimes \sigma_x ,&\rho(A_{25})=\tfrac12 \sigma_z \otimes \sigma_x\otimes \epsilon\otimes I ,\\ \rho(A_{12})= \tfrac12 I \otimes \sigma_z\otimes \sigma_x\otimes \epsilon,&\rho(A_{26})=\tfrac12 \sigma_z \otimes \sigma_x\otimes \sigma_z\otimes I ,\\ \rho(A_{13})= \tfrac12 I \otimes \sigma_z\otimes \sigma_x\otimes \sigma_z,&\rho(A_{27})=\tfrac12 \sigma_z \otimes \epsilon\otimes I\otimes I,\\ \rho(A_{14})= \tfrac12 I \otimes \sigma_z\otimes \epsilon\otimes I,&\rho(A_{28})=\tfrac12 \sigma_z \otimes \sigma_z\otimes I\otimes I. \end{array} \end{align} $$

Looking at the first factor in all of these generators, we observe that it is either I or $\sigma _z$ , (i.e., it is diagonal). This means that this 16-dimensional representation of $\mathbb {R}\oplus \mathfrak {so}(4,4)$ is reducible. It splits onto two real $8$ -dimensional Weyl representations

$$ \begin{align*}\rho=\rho_+\oplus\rho_-\quad\mathrm{in}\quad S=S_+\oplus S_-, \quad \mathrm{dim}_{\mathbb{R}} S_\pm=8\end{align*} $$

in the spaces $S_\pm $ of (Majorana)-Weyl spinors.

On generators of $\mathfrak {so}(4,4)$ , these two 8-dimensional representations $\rho _\pm $ are given by

(9.4) $$ \begin{align} \begin{array}{ll} \rho_\pm(A_1)= \tfrac12 I\otimes I\otimes \sigma_z,&\rho_\pm(A_{15})=\tfrac12 \sigma_z\otimes \sigma_z\otimes I,\\ \rho_\pm(A_2)= \tfrac12 I\otimes I\otimes \epsilon ,&\rho_\pm(A_{16})=\tfrac12 \epsilon\otimes \sigma_x\otimes \sigma_x,\\ \rho_\pm(A_3)= \tfrac12 I\otimes I\otimes \sigma_x ,&\rho_\pm(A_{17})=\tfrac12 \epsilon\otimes \sigma_x\otimes \epsilon ,\\ \rho_\pm(A_4)=\tfrac12 I\otimes \sigma_z\otimes \sigma_x,&\rho_\pm(A_{18})=\tfrac12 \epsilon\otimes \sigma_x\otimes \sigma_z ,\\ \rho_\pm(A_5)= \tfrac12 I\otimes \sigma_z\otimes \epsilon,&\rho_\pm(A_{19})=\tfrac12 \epsilon\otimes \epsilon\otimes I ,\\ \rho_\pm(A_6)= \tfrac12 I\otimes \sigma_z\otimes \sigma_z,&\rho_\pm(A_{20})=\tfrac12 \epsilon\otimes \sigma_z\otimes I ,\\ \rho_\pm(A_7)= \tfrac12 I\otimes \epsilon\otimes \sigma_x,&\rho_\pm(A_{21})=\tfrac12 \sigma_x\otimes I\otimes I,\\ \rho_\pm(A_8)= \tfrac12 I\otimes \epsilon\otimes \epsilon,&\rho_\pm(A_{22})=\pm\tfrac12 \sigma_x\otimes \sigma_x\otimes \sigma_x ,\\ \rho_\pm(A_9)= \tfrac12 I\otimes \epsilon\otimes \sigma_z,&\rho_\pm(A_{23})=\pm\tfrac12 \sigma_x\otimes \sigma_x\otimes \epsilon ,\\ \rho_\pm(A_{10})=\tfrac12 I\otimes \sigma_x\otimes I,&\rho_\pm(A_{24})=\pm\tfrac12 \sigma_x\otimes \sigma_x\otimes \sigma_z ,\\ \rho_\pm(A_{11})= \tfrac12 \sigma_z\otimes \sigma_x\otimes \sigma_x ,&\rho_\pm(A_{25})=\pm\tfrac12 \sigma_x\otimes \epsilon\otimes I ,\\ \rho_\pm(A_{12})= \tfrac12 \sigma_z\otimes \sigma_x\otimes \epsilon,&\rho_\pm(A_{26})=\pm\tfrac12 \sigma_x\otimes \sigma_z\otimes I ,\\ \rho_\pm(A_{13})= \tfrac12 \sigma_z\otimes \sigma_x\otimes \sigma_z,&\rho_\pm(A_{27})=\pm\tfrac12 \epsilon\otimes I\otimes I,\\ \rho_\pm(A_{14})= \tfrac12 \sigma_z\otimes \epsilon\otimes I,&\rho_\pm(A_{28})=\pm\tfrac12 \sigma_z\otimes I\otimes I. \end{array} \end{align} $$

We extend them to $\mathbb {R}\oplus \mathfrak {so}(4,4)$ by adding

$$ \begin{align*}\rho_\pm(A_{29})=\tfrac12 I\otimes I\otimes I.\end{align*} $$

It follows that the Weyl representations $(\rho _\pm ,S_\pm )$ of $\mathfrak {so}(4,4)$ are irreducible and nonequivalent.

They can be used to find yet another real 8-dimensional representation of $\mathfrak {so}(4,4)$ . For this, one considers the tensor product representation

$$ \begin{align*}\rho_+\otimes\rho_-.\end{align*} $$

This 64-dimensional real representation of $\mathfrak {so}(4,4)$ is reducible. It decomposes as

$$ \begin{align*}\rho_+\otimes\rho_-=\alpha\oplus\tau\quad\mathrm{in}\quad S_+\otimes S_-=T_{56}\oplus R,\quad\mathrm{with}\quad \dim_{\mathbb{R}}(R)=8,\,\,\dim_{\mathbb{R}}(T_{56})=56,\end{align*} $$

having irreducible components $(\alpha ,T_{56})$ and $(\tau ,R)$ of respective dimensions 56 and 8. Explicitly, on generators of $\mathbb {R}\oplus \mathfrak {so}(4,4)$ , the 8-dimensional representation $\tau $ reads

(9.5) $$ \begin{align}\begin{aligned} &\tau(A_1)=E_{66}-E_{22},\\ &\tau(A_2)=\tfrac12(E_{23}-E_{32}+E_{25}-E_{52}+E_{36}-E_{63}+E_{56}-E_{65}),\\ &\tau(A_3)=\tfrac12(E_{23}+E_{32}+E_{25}+E_{52}+E_{36}+E_{63}+E_{56}+E_{65}),\\ &\tau(A_4)=\tfrac12(E_{23}+E_{32}-E_{25}-E_{52}-E_{36}-E_{63}+E_{56}+E_{65}),\\ &\tau(A_5)=\tfrac12(E_{23}-E_{32}-E_{25}+E_{52}-E_{36}+E_{63}+E_{56}-E_{65}),\\ &\tau(A_6)=E_{33}-E_{55},\\ &\tau(A_7)=\tfrac12(E_{12}-E_{21}-E_{16}+E_{61}-E_{27}+E_{72}+E_{67}-E_{76}),\\ &\tau(A_8)=\tfrac12(-E_{12}-E_{21}-E_{16}-E_{61}-E_{27}-E_{72}-E_{67}-E_{76}),\\ &\tau(A_9)=\tfrac12(E_{13}-E_{31}+E_{15}-E_{51}-E_{37}+E_{73}-E_{57}+E_{75}),\\ &\tau(A_{10})=\tfrac12(E_{13}+E_{31}-E_{15}-E_{51}+E_{37}+E_{73}-E_{57}-E_{75}),\\ &\tau(A_{11})=\tfrac12(-E_{12}-E_{21}+E_{16}+E_{61}+E_{27}+E_{72}-E_{67}-E_{76}),\\ &\tau(A_{12})=\tfrac12(E_{12}-E_{21}+E_{16}-E_{61}+E_{27}-E_{72}+E_{67}-E_{76}),\\ &\tau(A_{13})=\tfrac12(-E_{13}-E_{31}-E_{15}-E_{51}+E_{37}+E_{73}+E_{57}+E_{75}),\\ &\tau(A_{14})=\tfrac12(-E_{13}+E_{31}+E_{15}-E_{51}-E_{37}+E_{73}+E_{57}-E_{75}),\\ &\tau(A_{15})=E_{11}-E_{77},\\ &\tau(A_{16})=\tfrac12(E_{24}-E_{42}+E_{28}-E_{82}+E_{46}-E_{64}-E_{68}+E_{86}),\\ &\tau(A_{17})=\tfrac12(E_{24}+E_{42}+E_{28}+E_{82}+E_{46}+E_{64}+E_{68}+E_{86}),\\ &\tau(A_{18})=\tfrac12(E_{34}-E_{43}+E_{38}-E_{83}-E_{45}+E_{54}+E_{58}-E_{85}),\\ &\tau(A_{19})=\tfrac12(-E_{34}-E_{43}-E_{38}-E_{83}+E_{45}+E_{54}+E_{58}+E_{85}),\\ &\tau(A_{20})=\tfrac12(-E_{14}+E_{41}-E_{18}+E_{81}+E_{47}-E_{74}-E_{78}+E_{87}),\\ &\tau(A_{21})=\tfrac12(E_{14}+E_{41}+E_{18}+E_{81}-E_{47}-E_{74}-E_{78}-E_{87}),\\ &\tau(A_{22})=\tfrac12(-E_{24}-E_{42}+E_{28}+E_{82}+E_{46}+E_{64}-E_{68}-E_{86}),\\ &\tau(A_{23})=\tfrac12(-E_{24}+E_{42}+E_{28}-E_{82}+E_{46}-E_{64}+E_{68}-E_{86}),\\ &\tau(A_{24})=\tfrac12(-E_{34}-E_{43}+E_{38}+E_{83}-E_{45}-E_{54}+E_{58}+E_{85}),\\ &\tau(A_{25})=\tfrac12(E_{34}-E_{43}-E_{38}+E_{83}+E_{45}-E_{54}+E_{58}-E_{85}),\\ &\tau(A_{26})=\tfrac12(E_{14}+E_{41}-E_{18}-E_{81}+E_{47}+E_{74}-E_{78}-E_{87}),\\ &\tau(A_{27})=\tfrac12(-E_{14}+E_{41}+E_{18}-E_{81}-E_{47}+E_{74}-E_{78}+E_{87}),\\ &\tau(A_{28})=-E_{44}+E_{88},\\ &\tau(A_{29})=E_{11}+E_{22}+E_{33}+E_{44}+E_{55}+E_{66}+E_{77}+E_{88}, \end{aligned}\end{align} $$

where $E_{ij}$ , $i,j=1,2,\dots ,8$ denote $8\times 8$ matrices with zeroes everywhere except the value 1 in the entry $(i,j)$ seating at the crossing of the ith row and the jth column.

The three real, irreducible, pairwise nonequivalent representations $(\rho _+,\rho _-,\tau )$ of $\mathfrak {so}(4,4)$ , given by the formulas (9.4) and (9.5), constitute the set of the triality representations for $\mathfrak {so}(4,4)$ .

9.2. Triality representations of $\mathfrak {so}(8,0)$

To get the real representation $(\rho ,S)$ of $\mathfrak {so}(8,0)$ in the space $S=\mathbb {R}^{16}$ of Dirac spinors, we need the real Dirac $\gamma $ matrices satisfying the Dirac identity (9.2), but now with

$$ \begin{align*}g_{ij}=\delta_{ij},\end{align*} $$

where $\delta _{ij}$ is the Kronecker delta in 8 dimensions.

Thus, we need to modify the Dirac matrices $\gamma _i$ from (9.1) to have the proper signature of the metric. This is done in two steps [Reference Trautman, Naber, Tsou and Françoise19]. First, one puts the imaginary unit i in front of some of the Dirac matrices $\gamma _i$ generating the Clifford algebra ${\mathcal C}\ell (4,4)$ to get the proper signature of $(g_{ij})$ . Although this produces few complex generators, in step two, one uses them with the others and modifies them in an algorithmic fashion so that they become all real and still satisfy the Dirac identity with the proper signature of $(g_{ij})$ . Explicitly, it is done as follows:

By placing the imaginary unit i in front of $\gamma _2$ , $\gamma _4$ , $\gamma _6$ and $\gamma _8$ in (9.1), we obtain 8 matrices

$$ \begin{align*}\tilde{\gamma}_{2j-1}=\gamma_{2j-1}, \quad \tilde{\gamma}_{2j}=i\gamma_{2j}, \quad j=1,2,3,4,\end{align*} $$

with $\gamma _i$ , $i,1,\dots ,8$ , in (9.1). These constitute generators of the complex 16-dimensional representation of the Clifford algebra ${\mathcal C}\ell (8,0)$ . We will also need the representation of this Clifford algebra, which is complex conjugate of $\tilde {\gamma }$ . This is generated by

$$ \begin{align*}\overline{\tilde{\gamma}}_{2j-1}=\gamma_{2j-1}, \quad \overline{\tilde{\gamma}}_{2j}=-\gamma_{2j}, \quad j=1,2,3,4.\end{align*} $$

The Clifford algebra representations generated by the Dirac matrices $\tilde {\gamma }$ and $\overline {\tilde {\gamma }}$ are real equivalent – that is, there exists a real $16\times 16$ matrix B such that

$$ \begin{align*}B\tilde{\gamma}_i=\overline{\tilde{\gamma}}_iB,\quad \forall i=1,\dots,8.\end{align*} $$

It can be chosen so that

$$ \begin{align*}B^2=\mathrm{Id},\end{align*} $$

where $\mathrm {Id}=I\otimes I\otimes I\otimes I$ .

Explicitly,

$$ \begin{align*}B=\sigma_z\otimes\epsilon\otimes\sigma_z\otimes \epsilon.\end{align*} $$

Using this matrix, we define a new set of eight $\gamma $ matricesFootnote ⁶ by

$$ \begin{align*}\gamma_i\,=\,(i B+\mathrm{Id})\,\,\tilde{\gamma}_i\,\,(i B+\mathrm{Id})^{-1}, \quad\quad\forall i=1,\dots,8.\end{align*} $$

One can check that these 8 matrices are all real and that they satisfy the desired Dirac identity:

$$ \begin{align*}\gamma_i\gamma_j+\gamma_j\gamma_i=2\delta_{ij} (I\otimes I\otimes I\otimes I), \quad i,j=1,\dots,8.\end{align*} $$

Explicitly, we have

$$ \begin{align*}\begin{aligned} &\gamma_1=\sigma_x\otimes\sigma_x\otimes\sigma_x\otimes\sigma_x\\ &\gamma_2=-\epsilon\otimes\sigma_z\otimes\epsilon\otimes I\\ &\gamma_3=\sigma_x\otimes\sigma_x\otimes\sigma_x\otimes\sigma_z\\ &\gamma_4=\epsilon\otimes\sigma_z\otimes\sigma_x\otimes \epsilon\\ &\gamma_5=\sigma_x\otimes\sigma_x\otimes\sigma_z\otimes I\\ &\gamma_6=-\epsilon\otimes I\otimes \sigma_z\otimes \epsilon\\ &\gamma_7=\sigma_x\otimes\sigma_z\otimes I\otimes I\\ &\gamma_8=\sigma_x\otimes \epsilon\otimes \sigma_z\otimes \epsilon. \end{aligned} \end{align*} $$

The 28 generators of $\mathfrak {so}(8,0)$ in the Majorana-Dirac spinor representation $\rho $ in the space of Dirac spinors $S=\mathbb {R}^{16}$ are given by

$$ \begin{align*}\rho(A_{I(i,j)})=\tfrac12\gamma_i\gamma_j, \quad 1\leq i<j\leq 8,\end{align*} $$

where again we have used the function $I=I(i,j)$ defined in (8.1). Note that since now $i<j$ can run from 1 to 8, the function has a range from 1 to 28. We add to this the scaling generator, $\rho (A_{29})$ , extending the Lie algebra $\mathfrak {so}(4,4)$ to $\mathfrak {coa}(4,4)$ , given by

$$ \begin{align*}\rho(A_{29})=\tfrac12 I\otimes I\otimes I\otimes I.\end{align*} $$

In terms of the 2-dimensional Pauli matrices, these generators look like

(9.6) $$ \begin{align} \begin{array}{ll} \rho(A_1)= -\tfrac12 \sigma_z \otimes \epsilon\otimes \sigma_z\otimes \sigma_x,&\rho(A_{15})=-\tfrac12 \sigma_z \otimes \sigma_x\otimes I\otimes \epsilon,\\ \rho(A_2)= \tfrac12 I \otimes I\otimes I\otimes \epsilon ,&\rho(A_{16})=\tfrac12 I \otimes \epsilon\otimes \sigma_x\otimes \sigma_x,\\ \rho(A_3)= \tfrac12 \sigma_z \otimes \epsilon\otimes \sigma_z\otimes \sigma_z ,&\rho(A_{17})=\tfrac12 \sigma_z \otimes I \otimes \epsilon\otimes I ,\\ \rho(A_4)=\tfrac12 \sigma_z \otimes \epsilon\otimes I\otimes \sigma_z,&\rho(A_{18})=\tfrac12 I \otimes \epsilon\otimes \sigma_x\otimes \sigma_z ,\\ \rho(A_5)= -\tfrac12 I \otimes I\otimes \sigma_z\otimes \epsilon,&\rho(A_{19})=-\tfrac12 \sigma_z \otimes I\otimes \sigma_x\otimes \epsilon ,\\ \rho(A_6)= -\tfrac12 \sigma_z \otimes \epsilon\otimes I\otimes \sigma_x,&\rho(A_{20})=\tfrac12 I \otimes \epsilon\otimes \sigma_z\otimes I ,\\ \rho(A_7)= \tfrac12 I \otimes I\otimes \epsilon\otimes \sigma_x,&\rho(A_{21})=\tfrac12 \sigma_z \otimes \sigma_z\otimes \sigma_z\otimes \epsilon,\\ \rho(A_8)= -\tfrac12 \sigma_z \otimes \epsilon\otimes \sigma_x\otimes I,&\rho(A_{22})=\tfrac12 I \otimes \sigma_z\otimes \epsilon\otimes \sigma_z ,\\ \rho(A_9)= \tfrac12 I \otimes I\otimes \epsilon\otimes \sigma_z,&\rho(A_{23})=-\tfrac12 \sigma_z \otimes \sigma_x\otimes \sigma_x\otimes \epsilon ,\\ \rho(A_{10})=\tfrac12 \sigma_z \otimes \epsilon\otimes \epsilon\otimes \epsilon,&\rho(A_{24})=-\tfrac12 I \otimes \sigma_z\otimes \epsilon\otimes \sigma_x ,\\ \rho(A_{11})= -\tfrac12 \sigma_z \otimes \sigma_x\otimes \epsilon\otimes \sigma_z ,&\rho(A_{25})=-\tfrac12 \sigma_z \otimes \sigma_x\otimes \epsilon\otimes I ,\\ \rho(A_{12})= -\tfrac12 I \otimes \sigma_z\otimes \sigma_x\otimes \epsilon,&\rho(A_{26})=\tfrac12 I \otimes \sigma_z\otimes I\otimes \epsilon ,\\ \rho(A_{13})=\tfrac12 \sigma_z \otimes \sigma_x\otimes \epsilon\otimes \sigma_x,&\rho(A_{27})=-\tfrac12 \sigma_z \otimes \epsilon\otimes I\otimes I,\\ \rho(A_{14})= -\tfrac12 I \otimes \sigma_z\otimes \epsilon\otimes I,&\rho(A_{28})=-\tfrac12 I \otimes \sigma_x\otimes \sigma_z\otimes \epsilon. \end{array} \end{align} $$

Similarly, as in the case of $\mathfrak {so}(4,4)$ , this 16-dimensional representation of $\mathbb {R}\oplus \mathfrak {so}(4,4)$ is reducible, again due to the appearance of I and $\sigma _z$ only as the first factors in the above formulas. It splits onto two real $8$ -dimensional Weyl representations

$$ \begin{align*}\rho=\rho_+\oplus\rho_-\quad\mathrm{in}\quad S=S_+\oplus S_-, \quad \mathrm{dim}_{\mathbb{R}} S_\pm=8.\end{align*} $$

On generators of $\mathfrak {so}(8,0)$ , these two 8-dimensional representations $\rho _\pm $ , are given by

(9.7) $$ \begin{align} \begin{array}{ll} \rho_\pm(A_1)= \mp\tfrac12 \epsilon\otimes \sigma_z\otimes \sigma_x,&\rho_\pm(A_{15})=\mp\tfrac12 \sigma_x\otimes I\otimes \epsilon,\\ \rho_\pm(A_2)= \tfrac12 I\otimes I\otimes \epsilon ,&\rho_\pm(A_{16})=\tfrac12 \epsilon\otimes \sigma_x\otimes \sigma_x,\\ \rho_\pm(A_3)= \pm\tfrac12 \epsilon\otimes \sigma_z\otimes \sigma_z ,&\rho_\pm(A_{17})=\pm\tfrac12 I \otimes \epsilon\otimes I ,\\ \rho_\pm(A_4)=\pm\tfrac12 \epsilon\otimes I\otimes \sigma_z,&\rho_\pm(A_{18})=\tfrac12 \epsilon\otimes \sigma_x\otimes \sigma_z ,\\ \rho_\pm(A_5)= -\tfrac12 I\otimes \sigma_z\otimes \epsilon,&\rho_\pm(A_{19})=\mp\tfrac12 I\otimes \sigma_x\otimes \epsilon ,\\ \rho_\pm(A_6)= \mp\tfrac12 \epsilon\otimes I\otimes \sigma_x,&\rho_\pm(A_{20})=\tfrac12 \epsilon\otimes \sigma_z\otimes I ,\\ \rho_\pm(A_7)= \tfrac12 I\otimes \epsilon\otimes \sigma_x,&\rho_\pm(A_{21})=\pm\tfrac12 \sigma_z\otimes \sigma_z\otimes \epsilon,\\ \rho_\pm(A_8)= \mp\tfrac12 \epsilon\otimes \sigma_x\otimes I,&\rho_\pm(A_{22})=\tfrac12 \sigma_z\otimes \epsilon\otimes \sigma_z ,\\ \rho_\pm(A_9)= \tfrac12 I\otimes \epsilon\otimes \sigma_z,&\rho_\pm(A_{23})=\mp\tfrac12 \sigma_x\otimes \sigma_x\otimes \epsilon ,\\ \rho_\pm(A_{10})=\pm\tfrac12 \epsilon\otimes \epsilon\otimes \epsilon,&\rho_\pm(A_{24})=-\tfrac12 \sigma_z\otimes \epsilon\otimes \sigma_x ,\\ \rho_\pm(A_{11})= \mp\tfrac12 \sigma_x\otimes \epsilon\otimes \sigma_z ,&\rho_\pm(A_{25})=\mp\tfrac12 \sigma_x\otimes \epsilon\otimes I ,\\ \rho_\pm(A_{12})= -\tfrac12 \sigma_z\otimes \sigma_x\otimes \epsilon,&\rho_\pm(A_{26})=\tfrac12 \sigma_z\otimes I\otimes \epsilon ,\\ \rho_\pm(A_{13})=\pm\tfrac12 \sigma_x\otimes \epsilon\otimes \sigma_x,&\rho_\pm(A_{27})=\mp\tfrac12 \epsilon\otimes I\otimes I,\\ \rho_\pm(A_{14})= -\tfrac12 \sigma_z\otimes \epsilon\otimes I,&\rho_\pm(A_{28})=-\tfrac12 \sigma_x\otimes \sigma_z\otimes \epsilon. \end{array} \end{align} $$

We extend them em to $\mathbb {R}\oplus \mathfrak {so}(8,0)$ by adding

$$ \begin{align*}\rho_\pm(A_{29})=\tfrac12 I\otimes I\otimes I.\end{align*} $$

It follows that the Weyl representations $(\rho _\pm ,S_\pm )$ of $\mathfrak {so}(4,4)$ are irreducible and nonequivalent.

We use them to find the defining representation $(\tau ,R)$ of $\mathfrak {so}(8,0)$ in the vector space $R=\mathbb {R}^8$ of vectors. We again consider the tensor product representation $\rho _+\otimes \rho _-$ . It decomposes as

$$ \begin{align*}\rho_+\otimes\rho_-=\alpha\oplus\tau\quad\mathrm{in}\quad S_+\otimes S_-=T_{56}\oplus R,\quad\mathrm{with}\quad \dim_{\mathbb{R}}(R)=8,\,\,\dim_{\mathbb{R}}(T_{56})=56,\end{align*} $$

having irreducible components $(\alpha ,T_{56})$ and $(\tau ,R)$ of respective dimensions 56 and 8. Explicitly, on generators $A_I$ of $\mathbb {R}\oplus \mathfrak {so}(8,0)$ , the 8-dimensional representation $\tau $ reads

(9.8) $$ \begin{align} \begin{array}{llll} \tau(A_1)=E_{38}-E_{83},&\quad \tau(A_8)=E_{35}-E_{53},&\quad \tau(A_{15})=E_{52}-E_{25},&\quad \tau(A_{22})=E_{68}-E_{86},\\ \tau(A_2)=E_{78}-E_{87}, &\quad\tau(A_9)=E_{75}-E_{57}, &\quad \tau(A_{16})=E_{18}-E_{81},&\quad \tau(A_{23})=E_{36}-E_{63}, \\ \tau(A_3)=E_{37}-E_{73}, &\quad\tau(A_{10})=E_{54}-E_{45},&\quad \tau(A_{17})=E_{31}-E_{13},&\quad \tau(A_{24})=E_{76}-E_{67},\\ \tau(A_4)=E_{84}-E_{48},&\quad \tau(A_{11})=E_{28}-E_{82},&\quad \tau(A_{18})=E_{71}-E_{17},&\quad \tau(A_{25})=E_{64}-E_{46},\\ \tau(A_5)=E_{43}-E_{34},&\quad \tau(A_{12})=E_{32}-E_{23},&\quad \tau(A_{19})=E_{14}-E_{41},&\quad \tau(A_{26})=E_{56}-E_{65},\\ \tau(A_6)=E_{47}-E_{74},&\quad \tau(A_{13})=E_{72}-E_{27},&\quad \tau(A_{20})=E_{51}-E_{15},&\quad \tau(A_{27})=E_{26}-E_{62},\\ \tau(A_7)=E_{58}-E_{85}, &\quad \tau(A_{14})= E_{24}-E_{42},&\quad \tau(A_{21})=E_{21}-E_{12},&\quad \tau(A_{28})=E_{16}-E_{61}, \end{array} \end{align} $$

where $E_{ij}$ , $i,j=1,2,\dots ,8$ denote $8\times 8$ matrices with zeroes everywhere except the value 1 in the entry $(i,j)$ seating at the crossing of the ith row and the jth column.

The three real, irreducible, pairwise nonequivalent representations $(\rho _+,\rho _-,\tau )$ of $\mathfrak {so}(8,0)$ given by the formulas (9.7) and (9.8) constitute the set of the triality representations for $\mathfrak {so}(8,0)$ .

10. Application: 2-step graded realisations of real forms of the exceptional Lie algebra $\mathfrak {e}_6$

The simple exceptional complex Lie algebra $\mathfrak {e}_6$ has the following noncompact real forms:

1. (1) $\mathfrak {e}_I$ , with Satake diagram ,

2. (2) $\mathfrak {e}_{II}$ , with Satake diagram ,

3. (3) $\mathfrak {e}_{III}$ , with Satake diagram , and

4. (4) $\mathfrak {e}_{IV}$ , with Satake diagram .

Èlie Cartan in his theses [Reference Cartan8, Reference Cartan9] mentioned realisation of the real form $\mathfrak {e}_I$ in $N=\mathbb {R}^{16}$ . In the modern language, Cartan’s realisation is such that $\mathfrak {e}_I$ is the algebra of authomorphisms of the flat model of a parabolic geometry of type $(E_I,P)$ , where the choice of parabolic subgroup in the real form $E_I$ of the exceptional Lie group $\mathbf { E}_6$ is indicated by the following decoration of the Satake diagram for $\mathfrak {e}_I$ : . The structure on the 16-dimensional manifold $N=E_I/P$ , whose symmetry is $E_I$ , is a Majorana-Weyl $\mathbb {R}\mathbf {Spin}(5,5)$ structure (i.e., the reduction of the structure group of the tangent bundle $\mathrm {T}N$ to the $\mathbb {R}\mathbf {Spin}(5,5)\subset \mathbf {GL}(16,\mathbb {R})$ in the irreducible 16-dimensional representation of Majorana-Weyl spinors [Reference Trautman, Naber, Tsou and Françoise19]). This geometry, as 1-step graded, is quite different from 2-step graded geometries considered in our paper. We also mention that if we wanted to have a realisation of, say $\mathfrak {e}_{II}$ or $\mathfrak {e}_{III}$ , in the spirit of Cartan’s realisation of $\mathfrak {e}_I$ (i.e., if we crossed one lateral node in the Satake diagram of $\mathfrak {e}_{II}$ or $\mathfrak {e}_{III}$ ), we would be forced to cross the complex conjugated lateral root, resulting in the Satake diagrams or , which would give realisations of the respective $\mathfrak {e}_{II}$ and $\mathfrak {e}_{III}$ in dimension twenty four. This we did in [Reference Hill, Merker, Nie and Nurowski14], providing realisations of $\mathfrak {e}_{II}$ and $\mathfrak {e}_{III}$ as Lie algebras of CR-authomorphisms of certain 24-dimensional CR manifolds of CR dimension 16, and CR (real) codimension 8. The important point of these realisations of these two real forms of $\mathfrak {e}_6$ was that these geometries were 2-step graded, as in the case of Cartan’s realisation of $\mathfrak {f}_I$ , and they could have been also thought as realisations in terms of the symmetry algebras of the structure $(M,{\mathcal D})$ , where M is a certain 24-dimensional real manifold and $\mathcal D$ is a real rank 16-distribution on M with $[{\mathcal D},{\mathcal D}]=\mathrm {T}M$ . Thus, these two geometries described by us in [Reference Hill, Merker, Nie and Nurowski14] are 2-step graded geometries of distributions – very much like Cartan’s realisation of $\mathfrak {f}_I$ .

In this section, we give the remaining similar realisations of the yet untreated cases of $\mathfrak {e}_{II}$ and $\mathfrak {e}_{III}$ .

10.1. Realisations of $\mathfrak {e}_{I}$ and $\mathfrak {e}_{IV}$ : generalities

To get realisations of $\mathfrak {e}_{I}$ and $\mathfrak {e}_{IV}$ in dimension 24, we decorate the Satake diagrams of these two Lie algebras as follows: and . These choices of a parabolic subalgebra in the respective $\mathfrak {e}_{I}$ and $\mathfrak {e}_{IV}$ produces the following gradation in these algebras:

with

being 2-step nilpotent and having grading components $\mathfrak {n}_{-2A}$ and of respective dimension $r_A=8$ and $s_A=16$ ,

The Lie algebra $\mathfrak {n}_{0A}$ in the Tanaka prolongation of up to $0^{th}$ order is

1. (1) $\mathfrak {n}_{0I}=2\mathbb {R}\oplus \mathfrak {so}(4,4)=\mathbb {R}\oplus \mathfrak {co}(4,4)$ in the case of $\mathfrak {e}_I$ , and

2. (2) $\mathfrak {n}_{0IV}=2\mathbb {R}\oplus \mathfrak {so}(8,0)=\mathbb {R}\oplus \mathfrak {co}(8,0)$ in the case of $\mathfrak {e}_{IV}$ .

The last two statements, (1) and (2), get clear when one looks at the Satake diagrams we have just decorated. If we strip off the crossed nodes from these diagrams, we get and , clearly the simple part of $\mathfrak {n}_{0A}$ s above.

Because of the grading property $[\mathfrak {n}_{iA},\mathfrak {n}_{jA}]\subset \mathfrak {n}_{(i+j)A}$ in the Lie algebras $\mathfrak {e}_A$ , restricting to subalgebras , we see that we have representations and given by the adjoint action of $\mathfrak {co}(4,4)$ or $\mathfrak {co}(8,0)$ which naturally seat in $\mathfrak {n}_{0A}$ , respectively.

There is no surprise that the representations are the Dirac spinor representations (9.3) and (9.6) of the respective $\mathfrak {co}(4,4)$ and $\mathfrak {co}(8,0)$ parts of $\mathfrak {n}_{0A}$ s in the 16-dimensional real vector spaces . As such, these representations are reducible, and they split each $\mathfrak {n}_{0A}$ , $A=I,IV$ onto two irreducible representations in real 8-dimensional spaces of Weyl spinors. This shows that the 2-step nilpotent Lie algebra is, for each $A=I,IV$ , a natural representation space for the action of the three triality representations $(\rho _+,\rho _-,\tau )$ . We have

and the 8-dimensional real irreducible representations $(\tau _A,\rho _A+,\rho _A-)$ of $\mathfrak {co}(4,4)$ or $\mathfrak {co}(8,0)$ acting in the respective , and .

We summarise the considerations from this section in the following theorem,

Theorem 10.1. (Natural realisation of the triality representations)

1. (1) The $\mathfrak {so}(4,4)$ triality: The real form $\mathfrak {e}_I$ of the simple exceptional Lie algebra $\mathfrak {e}_6$ , when graded according to the following decoration of its Satake diagram , has the part as a real 24-dimensional vector space, naturally split onto the three real 8-dimensional components , and ,

   

   This decomposition is $\mathfrak {so}(4,4)$ invariant and consists of components , and , on which the triality representation

   $$ \begin{align*}\tau\oplus\rho_+\oplus\rho_-\end{align*} $$
   of $\mathfrak {so}(4,4)$ acts irreducibly.

2. (2) The $\mathfrak {so}(8,0)$ triality: Likewise, the real form $\mathfrak {e}_{IV}$ of the simple exceptional Lie algebra $\mathfrak {e}_6$ , when graded according to the following decoration of its Satake diagram , has the part as a real 24-dimensional vector space, naturally split onto the three real 8-dimensional components , and ,

   
   This decomposition is $\mathfrak {so}(8,0)$ invariant and consists of components , and , on which the triality representation
   $$ \begin{align*}\tau\oplus\rho_+\oplus\rho_-\end{align*} $$
   of $\mathfrak {so}(8,0)$ acts irreducibly.

10.2. An explicit realisation of $\mathfrak {e}_I$ in dimension 24

Taking as $(\rho ,S)$ the Dirac spinors representation (9.3) of $\mathrm {co}(4,4)$ in dimension 16, and as $(\tau ,R)$ the vectorial representation (9.5) of $\mathrm {co}(4,4)$ in dimension 8, we again are in the situation of a missing $\omega \in \mathrm {Hom}(\bigwedge ^2S,R)$ from the triple $(\rho ,\tau ,\omega )$ described by the magical equation (2.1). Solving this equation for $\omega $ , we obtain $\omega ^i{}_{\mu \nu }$ , $i=1,\dots ,8$ , $\mu ,\nu =1,\dots ,16$ , which leads to the eight 2-forms $\omega ^i=\tfrac 12\omega ^i{}_{\mu \nu }\mathrm {d} x^\mu \wedge \mathrm {d} x^\nu $ on a 16-dimensional manifold $N=\mathbb {R}^{16}$ , which read

(10.1) $$ \begin{align}\begin{aligned} \omega^1=\,\,&-\mathrm{d} x^1\wedge\mathrm{d} x^{10}+\mathrm{d} x^2\wedge\mathrm{d} x^{9}+\mathrm{d} x^7\wedge\mathrm{d} x^{16}-\mathrm{d} x^8\wedge\mathrm{d} x^{15}\\ \omega^2=\,\,&-\mathrm{d} x^2\wedge\mathrm{d} x^{12}+\mathrm{d} x^4\wedge\mathrm{d} x^{10}+\mathrm{d} x^6\wedge\mathrm{d} x^{16}-\mathrm{d} x^8\wedge\mathrm{d} x^{14}\\ \omega^3=\,\,&-\mathrm{d} x^1\wedge\mathrm{d} x^{12}+\mathrm{d} x^4\wedge\mathrm{d} x^{9}+\mathrm{d} x^5\wedge\mathrm{d} x^{16}-\mathrm{d} x^8\wedge\mathrm{d} x^{13}\\ \omega^4=\,\,&-\mathrm{d} x^5\wedge\mathrm{d} x^{10}+\mathrm{d} x^6\wedge\mathrm{d} x^{9}+\mathrm{d} x^7\wedge\mathrm{d} x^{12}-\mathrm{d} x^8\wedge\mathrm{d} x^{11}\\ \omega^5=\,\,&-\mathrm{d} x^2\wedge\mathrm{d} x^{11}+\mathrm{d} x^3\wedge\mathrm{d} x^{10}+\mathrm{d} x^6\wedge\mathrm{d} x^{15}-\mathrm{d} x^7\wedge\mathrm{d} x^{14}\\ \omega^6=\,\,&-\mathrm{d} x^1\wedge\mathrm{d} x^{11}+\mathrm{d} x^3\wedge\mathrm{d} x^{9}+\mathrm{d} x^5\wedge\mathrm{d} x^{15}-\mathrm{d} x^7\wedge\mathrm{d} x^{13}\\ \omega^7=\,\,&-\mathrm{d} x^3\wedge\mathrm{d} x^{12}+\mathrm{d} x^4\wedge\mathrm{d} x^{11}+\mathrm{d} x^5\wedge\mathrm{d} x^{14}-\mathrm{d} x^6\wedge\mathrm{d} x^{13}\\ \omega^8=\,\,&-\mathrm{d} x^1\wedge\mathrm{d} x^{14}+\mathrm{d} x^2\wedge\mathrm{d} x^{13}+\mathrm{d} x^3\wedge\mathrm{d} x^{16}-\mathrm{d} x^4\wedge\mathrm{d} x^{15}. \end{aligned}\end{align} $$

The manifold $N=\mathbb {R}^{16}$ with these 2-forms, after contactification, gives the following Theorem.

Theorem 10.2. Let $M=\mathbb {R}^{24}$ with coordinates $(u^1,\dots ,u^8,x^1,\dots ,x^{16})$ , and consider eight 1-forms $\lambda ^1,\dots ,\lambda ^8$ on M given by

$$ \begin{align*}\begin{aligned} \lambda^1=\,\,&\mathrm{d} u^1-x^1\mathrm{d} x^{10}+ x^2\mathrm{d} x^{9}+ x^7\mathrm{d} x^{16}- x^8\mathrm{d} x^{15}\\ \lambda^2=\,\,&\mathrm{d} u^2- x^2\mathrm{d} x^{12}+ x^4\mathrm{d} x^{10}+ x^6\mathrm{d} x^{16}- x^8\mathrm{d} x^{14}\\ \lambda^3=\,\,&\mathrm{d} u^3- x^1\mathrm{d} x^{12}+ x^4\mathrm{d} x^{9}+x^5\mathrm{d} x^{16}- x^8\mathrm{d} x^{13}\\ \lambda^4=\,\,&\mathrm{d} u^4- x^5\mathrm{d} x^{10}+ x^6\mathrm{d} x^{9}+ x^7\mathrm{d} x^{12}- x^8\mathrm{d} x^{11}\\ \lambda^5=\,\,&\mathrm{d} u^5-x^2\mathrm{d} x^{11}+ x^3\mathrm{d} x^{10}+ x^6\mathrm{d} x^{15}- x^7\mathrm{d} x^{14}\\ \lambda^6=\,\,&\mathrm{d} u^6- x^1\mathrm{d} x^{11}+ x^3\mathrm{d} x^{9}+ x^5\mathrm{d} x^{15}- x^7\mathrm{d} x^{13}\\ \lambda^7=\,\,&\mathrm{d} u^7- x^3\mathrm{d} x^{12}+ x^4\mathrm{d} x^{11}+ x^5\mathrm{d} x^{14}- x^6\mathrm{d} x^{13}\\ \lambda^8=\,\,&\mathrm{d} u^8- x^1\mathrm{d} x^{14}+ x^2\mathrm{d} x^{13}+ x^3\mathrm{d} x^{16}-x^4\mathrm{d} x^{15}. \end{aligned}\end{align*} $$

The rank 16 distribution ${\mathcal D}$ on M defined as has its Lie algebra of infinitesimal authomorphisms $\mathfrak {aut}(\mathcal D)$ isomorphic to the Tanaka prolongation of , where $(\rho ,S=\mathbb {R}^{16})$ is the Dirac spinors representation (9.3) of $\mathfrak {n}_{00}=\mathfrak {co}(4,4)$ , and $(\tau ,R=\mathbb {R}^8)$ is the vectorial representation (9.5) of $\mathfrak {n}_{00}$ .

The symmetry algebra $\mathfrak {aut}({\mathcal D})$ is isomorphic to the simple exceptional Lie algebra $\mathfrak {e}_{I}$ ,

$$ \begin{align*}\mathfrak{aut}({\mathcal D})=\mathfrak{e}_{I},\end{align*} $$

having the following natural gradation

$$ \begin{align*}\mathfrak{aut}({\mathcal D})=\mathfrak{n}_{-2}\oplus\mathfrak{n}_{-1}\oplus\mathfrak{n}_0\oplus\mathfrak{n}_1\oplus\mathfrak{n}_2,\end{align*} $$

with $\mathfrak {n}_{-2}=R$ , $\mathfrak {n}_{-1}=S=S_+\oplus S_-$ ,

$$ \begin{align*}\mathfrak{n}_0=\mathbb{R}\oplus\mathfrak{co}(4,4)\supset \mathfrak{n}_{00},\end{align*} $$

$\mathfrak {n}_{1}=S^*$ , $\mathfrak {n}_{2}=R^*$ , and with the spaces $S_\pm $ being the carrier spaces for the Weyl spinors representations $\rho _\pm $ of $\mathfrak {co}(4,4)$ . The gradation in $\mathfrak {e}_I$ is inherited from the distribution structure $(M,{\mathcal D})$ . The duality signs $*$ at $R^*$ and $S^*$ above are with respect to the Killing form in $\mathfrak {e}_{I}$ .

The contactification $(M,{\mathcal D})$ is locally the flat model for the parabolic geometry of type $(E_{I},P_{I})$ related to the following crossed Satake diagram: .

Remark 10.3. Also the $\mathfrak {e}_I$ case, considered in this section, admits a description in terms of an $\mathbb {R}\oplus \mathfrak {so}(4,4)$ invariant 4-form $\Phi $ in $S=\mathbb {R}^{16}$ . Now $\Phi $ may be represented by

$$ \begin{align*}\Phi=h_{ij}\omega^i\wedge\omega^j,\end{align*} $$

where $\omega ^i$ are given by (10.1) and

$$ \begin{align*}\big(\,\,h_{ij}\,\,\big)\,\,= \,\,\tfrac12\,\,\begin{pmatrix} 0&0&0&0&0&0&1&0\\0&0&0&0&0&-1&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&0&0&1\\0&0&1&0&0&0&0&0\\0&-1&0&0&0&0&0&0\\1&0&0&0&0&0&0&0\\0&0&0&1&0&0&0&0 \end{pmatrix}.\end{align*} $$

Explicitly, the form $\Phi $ reads

(10.2) $$ \begin{align}\begin{aligned} \Phi\,\,=\,\,&2\mathrm{d} x^1\wedge\mathrm{d} x^2\wedge\mathrm{d} x^{11}\wedge\mathrm{d} x^{12}-2\mathrm{d} x^1\wedge\mathrm{d} x^3\wedge\mathrm{d} x^{10}\wedge\mathrm{d} x^{12}+\\ &2\mathrm{d} x^1\wedge\mathrm{d} x^4\wedge\mathrm{d} x^{10}\wedge\mathrm{d} x^{11}+2\mathrm{d} x^1\wedge\mathrm{d} x^5\wedge\mathrm{d} x^{10}\wedge\mathrm{d} x^{14}-\\&\mathrm{d} x^1\wedge\mathrm{d} x^6\wedge\mathrm{d} x^9\wedge\mathrm{d} x^{14}-\mathrm{d} x^1\wedge\mathrm{d} x^6\wedge\mathrm{d} x^{10}\wedge\mathrm{d} x^{13}-\\&\mathrm{d} x^1\wedge\mathrm{d} x^6\wedge\mathrm{d} x^{11}\wedge\mathrm{d} x^{16}+\mathrm{d} x^1\wedge\mathrm{d} x^6\wedge\mathrm{d} x^{12}\wedge\mathrm{d} x^{15}-\\ &2\mathrm{d} x^1\wedge\mathrm{d} x^7\wedge\mathrm{d} x^{12}\wedge\mathrm{d} x^{14}+2\mathrm{d} x^1\wedge\mathrm{d} x^8\wedge\mathrm{d} x^{11}\wedge\mathrm{d} x^{14}+\\&2\mathrm{d} x^2\wedge\mathrm{d} x^3\wedge\mathrm{d} x^9\wedge\mathrm{d} x^{12}-2\mathrm{d} x^2\wedge\mathrm{d} x^4\wedge\mathrm{d} x^9\wedge\mathrm{d} x^{11}-\\&\mathrm{d} x^2\wedge\mathrm{d} x^5\wedge\mathrm{d} x^9\wedge\mathrm{d} x^{14}-\mathrm{d} x^2\wedge\mathrm{d} x^5\wedge\mathrm{d} x^{10}\wedge\mathrm{d} x^{13}+\\ &\mathrm{d} x^2\wedge\mathrm{d} x^5\wedge\mathrm{d} x^{11}\wedge\mathrm{d} x^{16}-\mathrm{d} x^2\wedge\mathrm{d} x^5\wedge\mathrm{d} x^{12}\wedge\mathrm{d} x^{15}+\\ &2\mathrm{d} x^2\wedge\mathrm{d} x^6\wedge\mathrm{d} x^{9}\wedge\mathrm{d} x^{13}+2\mathrm{d} x^2\wedge\mathrm{d} x^7\wedge\mathrm{d} x^{12}\wedge\mathrm{d} x^{13}-\\&2\mathrm{d} x^2\wedge\mathrm{d} x^8\wedge\mathrm{d} x^{11}\wedge\mathrm{d} x^{13}+2\mathrm{d} x^3\wedge\mathrm{d} x^4\wedge\mathrm{d} x^{9}\wedge\mathrm{d} x^{10}-\\&2\mathrm{d} x^3\wedge\mathrm{d} x^5\wedge\mathrm{d} x^{10}\wedge\mathrm{d} x^{16}+2\mathrm{d} x^3\wedge\mathrm{ d} x^6\wedge\mathrm{d} x^{9}\wedge\mathrm{d} x^{16}+\\ &2\mathrm{d} x^3\wedge\mathrm{d} x^7\wedge\mathrm{d} x^{12}\wedge\mathrm{d} x^{16}-\mathrm{d} x^3\wedge\mathrm{d} x^8\wedge\mathrm{d} x^{9}\wedge\mathrm{d} x^{14}+\\&\mathrm{d} x^3\wedge\mathrm{d} x^8\wedge\mathrm{d} x^{10}\wedge\mathrm{d} x^{13}-\mathrm{d} x^3\wedge\mathrm{d} x^8\wedge\mathrm{d} x^{11}\wedge\mathrm{d} x^{16}-\\&\mathrm{d} x^3\wedge\mathrm{d} x^8\wedge\mathrm{d} x^{12}\wedge\mathrm{d} x^{15}+2\mathrm{d} x^4\wedge\mathrm{d} x^5\wedge\mathrm{d} x^{10}\wedge\mathrm{d} x^{15}-\\ &2\mathrm{d} x^4\wedge\mathrm{d} x^6\wedge\mathrm{d} x^{9}\wedge\mathrm{d} x^{15}+\mathrm{d} x^4\wedge\mathrm{d} x^7\wedge\mathrm{d} x^{9}\wedge\mathrm{d} x^{14}-\\ &\mathrm{d} x^4\wedge\mathrm{d} x^7\wedge\mathrm{d} x^{10}\wedge\mathrm{d} x^{13}-\mathrm{d} x^4\wedge\mathrm{d} x^7\wedge\mathrm{d} x^{11}\wedge\mathrm{d} x^{16}-\\&\mathrm{d} x^4\wedge\mathrm{d} x^7\wedge\mathrm{d} x^{12}\wedge\mathrm{d} x^{15}+2\mathrm{d} x^4\wedge\mathrm{d} x^8\wedge\mathrm{d} x^{11}\wedge\mathrm{d} x^{15}+\\&2\mathrm{d} x^5\wedge\mathrm{d} x^6\wedge\mathrm{d} x^{15}\wedge\mathrm{d} x^{16}-2\mathrm{d} x^5\wedge\mathrm{ d} x^7\wedge\mathrm{d} x^{14}\wedge\mathrm{d} x^{16}+\\ &2\mathrm{d} x^5\wedge\mathrm{d} x^8\wedge\mathrm{d} x^{14}\wedge\mathrm{d} x^{15}+2\mathrm{d} x^6\wedge\mathrm{d} x^7\wedge\mathrm{d} x^{13}\wedge\mathrm{d} x^{16}-\\&2\mathrm{d} x^6\wedge\mathrm{d} x^8\wedge\mathrm{d} x^{13}\wedge\mathrm{d} x^{15}+2\mathrm{d} x^7\wedge\mathrm{d} x^8\wedge\mathrm{d} x^{13}\wedge\mathrm{d} x^{14}. \end{aligned} \end{align} $$

This 4-form is such that its stabiliser in $\mathfrak {gl}(16,\mathbb {R})$ is $\mathfrak {n}_0=\mathbb {R}\oplus \mathfrak {co}(4,4)$ . When restricted to $\mathfrak {n}_{00}=\mathfrak {co}(4,4)$ this stabiliser is given precisely in the Mayorana Dirac spinor representation

$$ \begin{align*}\rho=\rho_+\oplus\rho_-\end{align*} $$

as in (9.3)–(9.4).

10.3. An explicit realisation of $\mathfrak {e}_{IV}$ in dimension 24

Similarly, as in the previous section, we take as $(\rho ,S)$ the Dirac spinors representation (9.6) of $\mathrm {co}(8.0)$ in dimension 16, and as $(\tau ,R)$ the vectorial representation (9.8) of $\mathrm {co}(8,0)$ in dimension 8, and we search for $\omega \in \mathrm {Hom}(\bigwedge ^2S,R)$ , solving the magical equation (2.1). We obtain $\omega ^i{}_{\mu \nu }$ , $i=1,\dots ,8$ , $\mu ,\nu =1,\dots ,16$ , which provides us with the eight 2-forms $\omega ^i=\tfrac 12\omega ^i{}_{\mu \nu }\mathrm {d} x^\mu \wedge \mathrm {d} x^\nu $ on a 16-dimensional manifold $N=\mathbb {R}^{16}$ , which read

(10.3) $$ \begin{align}\begin{aligned} \omega^1=\,\,&\mathrm{d} x^1\wedge\mathrm{d} x^{9}+\mathrm{d} x^2\wedge\mathrm{d} x^{10}+\mathrm{d} x^3\wedge\mathrm{d} x^{11}+\mathrm{d} x^4\wedge\mathrm{d} x^{12}-\\&\mathrm{d} x^5\wedge\mathrm{d} x^{13}-\mathrm{d} x^6\wedge\mathrm{d} x^{14}-\mathrm{d} x^7\wedge\mathrm{d} x^{15}-\mathrm{d} x^8\wedge\mathrm{d} x^{16}\\ \omega^2=\,\,&-\mathrm{d} x^1\wedge\mathrm{d} x^{10}+\mathrm{d} x^2\wedge\mathrm{d} x^{9}+\mathrm{d} x^3\wedge\mathrm{d} x^{12}-\mathrm{d} x^4\wedge\mathrm{d} x^{11}-\\&\mathrm{d} x^5\wedge\mathrm{d} x^{14}+\mathrm{d} x^6\wedge\mathrm{d} x^{13}+\mathrm{d} x^7\wedge\mathrm{d} x^{16}-\mathrm{d} x^8\wedge\mathrm{d} x^{15}\\ \omega^3=\,\,&-\mathrm{d} x^1\wedge\mathrm{d} x^{11}-\mathrm{d} x^2\wedge\mathrm{d} x^{12}+\mathrm{d} x^3\wedge\mathrm{d} x^{9}+\mathrm{d} x^4\wedge\mathrm{d} x^{10}+\\&\mathrm{d} x^5\wedge\mathrm{d} x^{15}+\mathrm{d} x^6\wedge\mathrm{d} x^{16}-\mathrm{d} x^7\wedge\mathrm{d} x^{13}-\mathrm{d} x^8\wedge\mathrm{d} x^{14}\\ \omega^4=\,\,&-\mathrm{d} x^1\wedge\mathrm{d} x^{12}+\mathrm{d} x^2\wedge\mathrm{d} x^{11}-\mathrm{d} x^3\wedge\mathrm{d} x^{10}+\mathrm{d} x^4\wedge\mathrm{d} x^{9}+\\&\mathrm{d} x^5\wedge\mathrm{d} x^{16}-\mathrm{d} x^6\wedge\mathrm{d} x^{15}+\mathrm{d} x^7\wedge\mathrm{d} x^{14}-\mathrm{d} x^8\wedge\mathrm{d} x^{13}\\ \omega^5=\,\,&\mathrm{d} x^1\wedge\mathrm{d} x^{13}+\mathrm{d} x^2\wedge\mathrm{d} x^{14}-\mathrm{d} x^3\wedge\mathrm{d} x^{15}-\mathrm{d} x^4\wedge\mathrm{d} x^{16}+\\&\mathrm{d} x^5\wedge\mathrm{d} x^{9}+\mathrm{d} x^6\wedge\mathrm{d} x^{10}-\mathrm{d} x^7\wedge\mathrm{d} x^{11}-\mathrm{d} x^8\wedge\mathrm{d} x^{12}\\ \omega^6=\,\,&\mathrm{d} x^1\wedge\mathrm{d} x^{14}-\mathrm{d} x^2\wedge\mathrm{d} x^{13}-\mathrm{d} x^3\wedge\mathrm{d} x^{16}+\mathrm{d} x^4\wedge\mathrm{d} x^{15}-\\&\mathrm{d} x^5\wedge\mathrm{d} x^{10}+\mathrm{d} x^6\wedge\mathrm{d} x^{9}+\mathrm{d} x^7\wedge\mathrm{d} x^{12}-\mathrm{d} x^8\wedge\mathrm{d} x^{11}\\ \omega^7=\,\,&\mathrm{d} x^1\wedge\mathrm{d} x^{15}-\mathrm{d} x^2\wedge\mathrm{d} x^{16}+\mathrm{d} x^3\wedge\mathrm{d} x^{13}-\mathrm{d} x^4\wedge\mathrm{d} x^{14}+\\&\mathrm{d} x^5\wedge\mathrm{d} x^{11}-\mathrm{d} x^6\wedge\mathrm{d} x^{12}+\mathrm{d} x^7\wedge\mathrm{d} x^{9}-\mathrm{d} x^8\wedge\mathrm{d} x^{10}\\ \omega^8=\,\,&-\mathrm{d} x^1\wedge\mathrm{d} x^{16}-\mathrm{d} x^2\wedge\mathrm{d} x^{15}-\mathrm{d} x^3\wedge\mathrm{d} x^{14}-\mathrm{d} x^4\wedge\mathrm{d} x^{13}-\\&\mathrm{d} x^5\wedge\mathrm{d} x^{12}-\mathrm{d} x^6\wedge\mathrm{d} x^{11}-\mathrm{d} x^7\wedge\mathrm{d} x^{10}-\mathrm{d} x^8\wedge\mathrm{d} x^{9}. \end{aligned}\end{align} $$

Contactifying, we have the following theorem:

Theorem 10.4. Let $M=\mathbb {R}^{24}$ with coordinates $(u^1,\dots ,u^8,x^1,\dots ,x^{16})$ , and consider eight 1-forms $\lambda ^1,\dots ,\lambda ^8$ on M given by

$$ \begin{align*}\begin{aligned} \lambda^1=\,\,&\mathrm{d} u^1+ x^1\mathrm{d} x^{9}+ x^2\mathrm{d} x^{10}+ x^3 \mathrm{d} x^{11}+ x^4 \mathrm{d} x^{12}- x^5 \mathrm{d} x^{13}- x^6 \mathrm{d} x^{14}- x^7 \mathrm{d} x^{15}- x^8 \mathrm{d} x^{16}\\ \lambda^2=\,\,&\mathrm{d} u^2- x^1 \mathrm{d} x^{10}+ x^2 \mathrm{d} x^{9}+ x^3 \mathrm{d} x^{12}- x^4 \mathrm{d} x^{11}- x^5 \mathrm{d} x^{14}+ x^6 \mathrm{d} x^{13}+ x^7 \mathrm{d} x^{16}- x^8 \mathrm{d} x^{15}\\ \lambda^3=\,\,&\mathrm{d} u^3- x^1 \mathrm{d} x^{11}- x^2 \mathrm{d} x^{12}+ x^3 \mathrm{d} x^{9}+ x^4 \mathrm{d} x^{10}+ x^5 \mathrm{d} x^{15}+ x^6 \mathrm{d} x^{16}- x^7 \mathrm{d} x^{13}- x^8 \mathrm{d} x^{14}\\ \lambda^4=\,\,&\mathrm{d} u^4- x^1 \mathrm{d} x^{12}+ x^2 \mathrm{d} x^{11}- x^3 \mathrm{d} x^{10}+ x^4 \mathrm{d} x^{9}+ x^5 \mathrm{d} x^{16}- x^6 \mathrm{d} x^{15}+ x^7 \mathrm{d} x^{14}- x^8 \mathrm{d} x^{13}\\ \lambda^5=\,\,&\mathrm{d} u^5+ x^1 \mathrm{d} x^{13}+ x^2 \mathrm{d} x^{14}- x^3 \mathrm{d} x^{15}- x^4 \mathrm{d} x^{16}+ x^5 \mathrm{d} x^{9}+ x^6 \mathrm{d} x^{10}- x^7 \mathrm{d} x^{11}- x^8 \mathrm{d} x^{12}\\ \lambda^6=\,\,&\mathrm{d} u^6+ x^1 \mathrm{d} x^{14}- x^2 \mathrm{d} x^{13}- x^3 \mathrm{d} x^{16}+ x^4 \mathrm{d} x^{15}- x^5 \mathrm{d} x^{10}+ x^6 \mathrm{d} x^{9}+ x^7 \mathrm{d} x^{12}- x^8 \mathrm{d} x^{11}\\ \lambda^7=\,\,&\mathrm{d} u^7+ x^1 \mathrm{d} x^{15}- x^2 \mathrm{d} x^{16}+ x^3 \mathrm{d} x^{13}- x^4 \mathrm{d} x^{14}+ x^5 \mathrm{d} x^{11}- x^6 \mathrm{d} x^{12}+ x^7 \mathrm{d} x^{9}- x^8 \mathrm{d} x^{10}\\ \lambda^8=\,\,&\mathrm{d} u^8 - x^1 \mathrm{d} x^{16}- x^2 \mathrm{d} x^{15}- x^3 \mathrm{d} x^{14}- x^4 \mathrm{d} x^{13}- x^5 \mathrm{d} x^{12}- x^6 \mathrm{d} x^{11}- x^7 \mathrm{d} x^{10}- x^8 \mathrm{d} x^{9}. \end{aligned}\end{align*} $$

The rank 16 distribution ${\mathcal D}$ on M defined as has its Lie algebra of infinitesimal authomorphisms $\mathfrak {aut}(\mathcal D)$ isomorphic to the Tanaka prolongation of , where $(\rho ,S=\mathbb {R}^{16})$ is the Dirac spinors representation (9.3) of $\mathfrak {n}_{00}=\mathfrak {co}(8,0)$ , and $(\tau ,R=\mathbb {R}^8)$ is the vectorial representation (9.5) of $\mathfrak {n}_{00}$ .

The symmetry algebra $\mathfrak {aut}({\mathcal D})$ is isomorphic to the simple exceptional Lie algebra $\mathfrak {e}_{IV}$ ,

$$ \begin{align*}\mathfrak{aut}({\mathcal D})=\mathfrak{e}_{IV},\end{align*} $$

having the following natural gradation:

$$ \begin{align*}\mathfrak{aut}({\mathcal D})=\mathfrak{n}_{-2}\oplus\mathfrak{n}_{-1}\oplus\mathfrak{n}_0\oplus\mathfrak{n}_1\oplus\mathfrak{n}_2,\end{align*} $$

with $\mathfrak {n}_{-2}=R$ , $\mathfrak {n}_{-1}=S=S_+\oplus S_-$ ,

$$ \begin{align*}\mathfrak{n}_0=\mathbb{R}\oplus\mathfrak{co}(8,0)\supset \mathfrak{n}_{00},\end{align*} $$

$\mathfrak {n}_{1}=S^*$ , $\mathfrak {n}_{2}=R^*$ , and with the spaces $S_\pm $ being the Carrier spaces for the Weyl spinors representations $\rho _\pm $ of $\mathfrak {co}(8,0)$ . The gradation in $e_{iV}$ is inherited from the distribution structure $(M,{\mathcal D})$ . The duality signs $*$ at $R^*$ and $S^*$ above are with respect to the Killing form in $\mathfrak {e}_{IV}$ .

The contactification $(M,{\mathcal D})$ is locally the flat model for the parabolic geometry of type $(E_{IV},P_{IV})$ related to the following crossed Satake diagram: .

Remark 10.5. Again, we have a description of the relevant representations in terms of an $\mathbb {R}\oplus \mathfrak {co}(8,0)$ invariant 4-form $\Phi $ in $S=\mathbb {R}^{16}$ . Now $\Phi $ may be represented by

$$ \begin{align*}\Phi=h_{ij}\omega^i\wedge\omega^j,\end{align*} $$

where $\omega ^i$ are given by (10.3) and

$$ \begin{align*}\big(\,\,h_{ij}\,\,\big)\,\,=\,\, \begin{pmatrix} 1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&1 \end{pmatrix}. \end{align*} $$

This 4-form is such that its stabiliser in $\mathfrak {gl}(16,\mathbb {R})$ is $\mathfrak {n}_0=\mathbb {R}\oplus \mathfrak {co}(8,0)$ . When restricted to $\mathfrak {n}_{00}=\mathfrak {co}(8,0)$ , this stabiliser is given precisely in the Mayorana Dirac spinor representation

$$ \begin{align*}\rho=\rho_+\oplus\rho_-\end{align*} $$

as in (9.6)–(9.7).

11. Application: one more realisation of $\mathfrak {e}_6$ and a realisation of $\mathfrak {b}_6$

Between the 24-dimensional realisations of $\mathfrak {e}_6$ mentioned in this paper, and Cartan’s 16-dimensional realisation of $\mathfrak {e}_6$ associated with the grading , there are 21-dimensional realisations of this algebra $\mathfrak {e}_6$ associated with the following Dynkin diagram crossing: . These define contact $\mathfrak {e}_6$ geometries and are described in [Reference Čap and Slovák11] p. 425–426.