= Spinor curvature identity
{title2=$[\nabla_\mu,\nabla_\nu]\psi=\tfrac14R_{\mu\nu AB}\gamma^{AB}\psi$}
The <spinor covariant derivative> has curvature $d\Omega+\Omega\wedge\Omega=\tfrac14R_{AB}\gamma^{AB}$, 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, $\nabla_{[\mu}\nabla_{\nu]}\psi=\tfrac18R_{\mu\nu AB}\gamma^{AB}\psi=\tfrac1{16}R_{\mu\nu AB}[\gamma^A,\gamma^B]\psi$. A noncoordinate frame requires its frame-commutator correction when using the operator commutator; the full second covariant derivative includes that correction automatically.
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