  In many paradigmatic materials, like Transition Metal Dichalcogenides, the
role played by the spin degrees of freedom is as important as the one played by
the electron-electron interaction. Thus an accurate treatment of the two
effects and of their interaction is necessary for an accurate and predictive
study of the optical and electronic properties of these materials. Despite the
GW-BSE approach correctly accounts for electronic correlations the spin-orbit
coupling effect is often neglected or treated perturbatively. Recently
spinorial formulations of GW-BSE have become available in different flavours in
material-science codes. Still an accurate validation and comparison of
different approaches is missing. In this work we go through the derivation of
non collinear GW-BSE approach. The scheme is applied to transition metal
dichalcogenides comparing perturbative and full spinorial approach. Our
calculations reveal that dark-bright exciton splittings are generally improved
when the spin orbit coupling is included non perturbatively. The
exchange-driven intravalley mixing between the A and B exciton is found to be
extremely important in the case of MoSe$_2$. We finally define the excitonic
spin and use it to sharply analyze the spinorial properties of Transition Metal
Dichalcogenides excitonic states.
