In local coordinates the Ricci identity reads

\[ T_{i_1\ldots i_k;pq}-T_{i_1\ldots i_k;qp} =R_{pqi_1}{}^mT_{mi_2\ldots i_k}+\cdots +R_{pqi_k}{}^mT_{i_1\ldots i_{k-1}m} \]

[1, Theorem 7.14].

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  1. J. Lee, Introduction to Riemannian Manifolds. Cham: Springer International Publishing, 2018. doi:10.1007/978-3-319-91755-9