Chwiedziuk et al. [‘No proper generalized quadratic forms are universal over quadratic fields’, Ramanujan J. 69 (2026), Article no. 92] recently proved that over a real quadratic field, a totally positive definite universal generalised quadratic form must contain a universal quadratic subform and asked whether this fails in higher degree. We prove it never does. We establish a sharper statement, valid over every totally real field: if such a form represents a value using a nonzero proper variable, that value is bounded below in at least two Archimedean embeddings. Universality follows at once, since a target can be made small in $d-1$ of the $d$ embeddings simultaneously, the unit rank of a totally real field of degree $d$ being $d-1$. The bound two is sharp. We analyse examples to explain the mechanisms governing the result in degrees $2$, $3$ and $4$.

The lifting problem for universal quadratic forms over a totally real number field K consists of determining the existence or otherwise of a quadratic form with integer coefficients (or $\mathbb {Z}$-form) that is universal over K. We prove the nonexistence of universal $\mathbb {Z}$-forms over simplest cubic fields for which the integer parameter is big enough. The monogenic case is already known. We prove the nonexistence in the nonmonogenic case by using the existence of a totally positive nonunit algebraic integer in K with minimal (codifferent) trace equal to one.