= Curvature-sign convention in Killing derivative identities
{title2=$k_{a;bc}=-C_{abc}{}^d k_d$}
Let $C^a{}_{bcd}=\partial_d\Gamma^a_{bc}-\partial_c\Gamma^a_{bd}-\Gamma^a_{ec}\Gamma^e_{bd}+\Gamma^a_{ed}\Gamma^e_{bc}$, with all other index positions obtained by the <metric tensor>. This convention is the negative of the convention $[\nabla_c,\nabla_d]V^a=R^a{}_{bcd}V^b$. The <Ricci identity> is $k_{a;bc}-k_{a;cb}=C_a{}^d{}_{bc}k_d$, with $k_{a;bc}=\nabla_c\nabla_b k_a$. For a <Killing vector>, the <Killing equation> makes this second derivative antisymmetric in $a,b$. Applying the commutator three times gives $2k_{a;bc}=(C_a{}^d{}_{bc}+C_c{}^d{}_{ba}-C_b{}^d{}_{ac})k_d=-2C_{abc}{}^d k_d$, where the last step uses the <first Bianchi identity>. Switching the sign of the <Riemann curvature tensor> switches the sign in this identity; moving a raised index by ordinary metric contraction does not switch it.
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