Second covariant derivative of a Killing vector
= Second covariant derivative of a Killing vector
For a <Killing vector field> and the curvature convention $[\nabla_\mu,\nabla_\nu]V^\alpha=R^\alpha{}_{\beta\mu\nu}V^\beta$,
$$
\nabla_\nu\nabla_\mu K^\alpha=R^\alpha{}_{\mu\nu\beta}K^\beta,
\qquad
\nabla_\mu\nabla^\mu K^\alpha=-R^\alpha{}_{\beta}K^\beta.
$$
The identity follows by combining the <Ricci identity>, <Killing equation>, and <first Bianchi identity>.