= Solution
Use signature $(-,+,+,+)$ and the <Riemann curvature tensor> convention $[\nabla_a,\nabla_b]X^c=R^c{}_{dab}X^d$. Lower the index of the <Killing vector field> and set $T_{abc}=\nabla_a\nabla_bV_c$. Differentiating the <Killing equation> gives $T_{abc}=-T_{acb}$, while the <Ricci identity> gives
$$
T_{abc}-T_{bac}=-R^d{}_{cab}V_d.
$$
Define $S_{abc}=R_{cbad}V^d$. The first-pair antisymmetry of the <Riemann curvature tensor> gives $S_{abc}=-S_{acb}$. The pair symmetry and the <first Bianchi identity> give
$$
S_{abc}-S_{bac}=(R_{cbad}-R_{cabd})V^d=-R_{dcab}V^d.
$$
Thus $D=T-S$ is symmetric in its first two indices and antisymmetric in its last two. These two symmetries force it to vanish:
$$
D_{abc}=D_{bac}=-D_{bca}=-D_{cba}=D_{cab}=D_{acb}=-D_{abc}.
$$
Consequently $T_{abc}=R_{cbad}V^d$. Raising $c$ using <metric compatibility> proves the required <second covariant derivative of a Killing vector>:
$$
\boxed{\nabla_a\nabla_bV^c=R^c{}_{bad}V^d.}
$$
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