= Solution
Expanding the two <covariant derivatives> of a vector, the second partial derivatives cancel in the commutator. The remaining derivatives and products of Christoffel symbols combine into the coordinate definition of the <Riemann curvature tensor>, giving the <Ricci identity>
$$
\boxed{2\nabla_{[\alpha}\nabla_{\beta]}V^\gamma
=[\nabla_\alpha,\nabla_\beta]V^\gamma
=R^\gamma{}_{\rho\alpha\beta}V^\rho.}
$$
For a type-$(2,0)$ tensor, the connection acts on both upper indices, so the <curvature commutator on a contravariant tensor> is
$$
\boxed{2\nabla_{[\alpha}\nabla_{\beta]}T^{\mu\nu}
=R^\mu{}_{\rho\alpha\beta}T^{\rho\nu}
+R^\nu{}_{\rho\alpha\beta}T^{\mu\rho}.}
$$
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