 Spinor curvature identity

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.