Einstein tensor contraction of spin curvature (${\gamma }^{\mu \nu \rho }[{\nabla }_{\nu },{\nabla }_{\rho }]=G^{\mu }{}_{\lambda }{\gamma }^{\lambda }$)

For the torsion-free spin connection with $[{\nabla }_{\nu },{\nabla }_{\rho }]=R_{\nu \rho ab}{\gamma }^{ab}/4$, Clifford expansion and the first Bianchi identity give ${\gamma }^{\mu \nu \rho }{R}_{\nu \rho ab}{\gamma }^{ab}=4R^{\mu }{}_{\lambda }{\gamma }^{\lambda }-2R{\gamma }^{\mu }$. Thus the antisymmetric double derivative equals $G^{\mu }{}_{\lambda }{\gamma }^{\lambda }/2$. This is the shared algebra behind quadratic-order minimal four-dimensional supergravity invariance and the Nester two-form divergence.