 Killing derivative contraction identity

For a Killing vector field, set $F_{ab}={\nabla }_a{K}_b$ and $a^s=K^t{\nabla }_t{K}^s$. The Killing equation makes $F$ antisymmetric, and expansion of normalized three-index antisymmetrization gives the displayed identity. On a Killing horizon, hypersurface orthogonality removes its first right-hand term and relates $F^{ab}{F}_{ab}$ to surface gravity.