Killing-spinor integrability with rescaled gamma matrices ($[D_a,D_b]=R_{abcd}{\gamma }^{cd}/(4s)+2c^2{\gamma }_{ab}$)

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.