= Riemann tensor Clifford contraction
{c}
{title2=$R_{abcd}\gamma^a\gamma^b\gamma^c\gamma^d=-2R\mathbf1$}
The four-gamma product decomposes into fully antisymmetric, two-gamma and scalar parts. The first part vanishes on contraction by the <first Bianchi identity>, and the two-gamma parts vanish because they contract symmetric <Ricci curvature> against antisymmetric gamma matrices. The scalar contractions are $-g^{ac}g^{bd}R_{abcd}+g^{ad}g^{bc}R_{abcd}=-2R$. Here $R=g^{ac}g^{bd}R_{abcd}$ fixes the curvature-slot convention.
// Target: second-fundamental-form.bigb
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