In the paper "The effects of interactions on the topological classification of free fermion systems", the authors demonstrate the existence of a quartic interaction W involving the 8 majorana operators $c_1 \ldots c_8$ (eq. 8) which is invariant under an SO(7) symmetry that can produce a unique ground state invariant under SO(7) as well as time-reversal. They also demonstrate that an SO(8) invariant interaction, which they call V does not work. I want to know the precise form of V which is not explicitly stated in their paper. The only thing the authors say is that V is also quartic in the fermion operators (above eq. 15) and is related to the quadratic Casimir.
I am confused because the 16 dimensional Hilbert space transforms as the spinor representation of SO(8) and is reducible into the two chiral irreps ($8_+$ and $8_-$). From Schur's lemma, the only SO(8) invariant term that can be written is proportional to the projector onto the two 8 dimensional irreps. Since the two irreps are distinguished by the eigenvalue of the chirality operator ($\Gamma_c \sim c_1 c_2 \ldots c_8$), I would think that the term V is proportional to $\Gamma_c$. But this is clearly not quartic.