= Chiral spinor curvature operator
{title2=$[\nabla_{AA'},\nabla_{BB'}]=\epsilon_{A'B'}\Delta_{AB}+\epsilon_{AB}\Delta_{A'B'}$}
The <spinor covariant derivative> commutator is a <differential two-form> in its derivative indices, so the <two-component spinor calculus> decomposition defines symmetric curvature operators $\Delta_{AB}$ and $\Delta_{A'B'}$. With $\epsilon^{AB}\epsilon_{AB}=2$, the unprimed projection is $\Delta_{AB}=\epsilon^{A'B'}[\nabla_{AA'},\nabla_{BB'}]/2$, and the primed projection is analogous. For a <torsion-free connection> these are algebraic curvature derivations: they annihilate scalar functions and obey the <Leibniz rule> on tensor products.
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