 Chiral Riemann curvature spinor decomposition

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 ${ϵ}^{AB}{ϵ}_{AB}=2$, its scalar is $\Lambda =X_{ABCD}{ϵ}^{AC}{ϵ}^{BD}/3$. This coefficient follows by contracting the displayed scalar term; different normalizations of that term give different scalar conventions.