= Chiral Riemann curvature spinor decomposition
{title2=$X_{ABCD}=\Psi_{ABCD}-\Lambda\epsilon_{D(A}\epsilon_{B)C}$}
Metric compatibility makes the curvature action on a two-component spinor symplectic, so its lowered curvature spinor is symmetric in each pair. The pair-exchange symmetry of the <Riemann curvature tensor> makes $X_{ABCD}=X_{CDAB}$. This six-component tensor decomposes into the five-component totally symmetric <Weyl curvature spinor> $\Psi_{ABCD}=X_{(ABCD)}$ and one scalar term. In the displayed normalization, with alternating forms satisfying $\epsilon^{AB}\epsilon_{AB}=2$, its scalar is $\Lambda=X_{ABCD}\epsilon^{AC}\epsilon^{BD}/3$. This coefficient follows by contracting the displayed scalar term; different normalizations of that term give different scalar conventions.
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