= Bianchi identities from covariant derivative commutators
The <first Bianchi identity> follows by applying cyclic curvature commutators to the gradient of an arbitrary <scalar field>: its symmetric Hessian makes the third-derivative terms cancel, leaving $R_{abc}{}^d+R_{bca}{}^d+R_{cab}{}^d=0$. For the <second Bianchi identity>, write the vector-bundle derivative as $D_a=\partial_a+\Gamma_a$, so $[D_b,D_c]=\mathcal R_{bc}$. The operator <Jacobi identity> gives the cyclic sum of $\partial_a\mathcal R_{bc}+[\Gamma_a,\mathcal R_{bc}]$ equal to zero. Restoring the connection terms on the two curvature-form indices gives precisely $\nabla_aR_{bcd}{}^e+\nabla_bR_{cad}{}^e+\nabla_cR_{abd}{}^e=0$: the additional terms cancel because the connection is torsion-free and curvature is antisymmetric in its first pair.
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