Spinor curvature identity ($[{\nabla }_{\mu },{\nabla }_{\nu }]\psi =\frac{1}{4}{R}_{\mu \nu AB}{\gamma }^{AB}\psi$)

The spinor covariant derivative has curvature $d\Omega +\Omega \wedge \Omega =\frac{1}{4}{R}_{AB}{\gamma }^{AB}$, since the spin representation preserves the Lorentz algebra bracket. In a coordinate basis with the torsion-free connection, antisymmetric second covariant derivatives therefore give the displayed commutator. With unit-weight antisymmetrization, ${\nabla }_{[\mu }{\nabla }_{\nu ]}\psi =\frac{1}{8}{R}_{\mu \nu AB}{\gamma }^{AB}\psi =\frac{1}{16}{R}_{\mu \nu AB}[{\gamma }^A,{\gamma }^B]\psi$. A noncoordinate frame requires its frame-commutator correction when using the operator commutator; the full second covariant derivative includes that correction automatically.

For $\{{\gamma }_a,{\gamma }_b\}=2s{g}_{ab}$, the compatible spinor covariant derivative has curvature $R_{abcd}{\gamma }^{cd}/(4s)$. The torsion-free connection preserves Clifford multiplication, so the mixed derivative terms cancel for $D_a={\nabla }_a+c{\gamma }_a$. Its integrability is therefore $(R_{abcd}{\gamma }^{cd}+8s{c}^2{\gamma }_{ab})ϵ=0$. Changing antisymmetrization weight cannot change the relative coefficient. The common normalization uses $s=1$; $s=2$ gives the coefficient $16c^2$.

Four independent complex Dirac solutions in four dimensions span each spinor fibre, because a connection-parallel solution is determined by its value at one point. Their integrability operator must therefore vanish on the whole fibre. Independence of the six Clifford bivectors then gives the displayed constant-curvature tensor and $R_{ab}=-12s{c}^2{g}_{ab}$. When $c>0$, the Ricci tensor is automatically nondegenerate.

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.