= Solution
Using the Leibniz rule for the connection, or first checking decomposable tensors and extending linearly, the commutator acts on each covariant index:
$$
[\nabla_\alpha,\nabla_\beta]W_{\mu\nu}
=-R^\rho{}_{\mu\alpha\beta}W_{\rho\nu}
-R^\rho{}_{\nu\alpha\beta}W_{\mu\rho}.
$$
Choose $W_{\mu\nu}=g_{\mu\nu}$. Metric compatibility makes the left side zero. Applying the formula with commutator indices $\beta,\gamma$ gives
$$
0=-R^\rho{}_{\mu\beta\gamma}g_{\rho\alpha}
-R^\rho{}_{\alpha\beta\gamma}g_{\mu\rho}
=-R_{\alpha\mu\beta\gamma}-R_{\mu\alpha\beta\gamma}.
$$
Hence
$$
\boxed{R_{\mu\alpha\beta\gamma}
=-R_{\alpha\mu\beta\gamma}.}
$$
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