This chapter revisits the construction of the Bethe wave function for the Lieb–Liniger model with spin degrees of freedom, which has traditionally followed a two-step approach: independent determination of coordinate and spin wave functions, later combined to enforce correct particle statistics. While this procedure is effective under the assumption that the hamiltonian is spin-independent, it becomes inadequate when spin-dependent interactions are present. Two key challenges arise: the need for a complete wave function without splitting it into separate coordinate and spin components, and the treatment of integrable systems with spin-dependent hamiltonians. To address these, the chapter introduces an efficient algebraic method based on the Faddeev–Zamolodchikov algebra, which provides a unified framework for handling coordinate and spin variables in integrable models with factorized scattering. This algebraic approach enables direct construction of the full wave function, offering a natural extension to systems involving both bosonic and fermionic degrees of freedom. The resulting spin chain representation of the transfer matrix not only simplifies the analysis but also yields a deeper understanding of the underlying algebraic structure governing these quantum systems.