Maximal Killing spinors force constant negative curvature ($R_{abcd}=-4s{c}^2(g_{ac}{g}_{bd}-g_{ad}{g}_{bc})$)

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.