The second Bianchi Identity is $$ \nabla_{[a}R_{bc]de}=0 $$
As far as I know, the proof (say, Wolfram MathWorld) starts by stating the representation of Riemann tensor in local inertial coordinates $$ R_{abcd}=\frac{1}{2}(\partial_a\partial_cg_{bd}-\partial_a\partial_dg_{bc}-\partial_b\partial_cg_{ad}+\partial_b\partial_dg_{ac}). $$
Then we calculate $$ \partial_aR_{bcde} $$
accordingly, and we say that it's true in a local inertial coordinate, and after changing the partial derivative into a covariant derivative, it's true in general.
My concern is, that I think we cannot express the Riemann tensor and the covariant derivative into local frames one by one, but should simultaneously. Say $$ \nabla_{a}R_{bcde}=\frac{1}{2}(\partial_a+\Gamma_1)(\partial_a\partial_cg_{bd}-\partial_a\partial_dg_{bc}-\partial_b\partial_cg_{ad}+\partial_b\partial_dg_{ac}+\Gamma_2) $$
where $\Gamma_1$ and $\Gamma_2$ are some terms involving the Christoffel symbol. When we only concern $R_{bcde}$ in a local frame, $\Gamma_2$ vanishes. But now we have a new term $$ \partial_a\Gamma_2 $$
which I cannot see vanish because it involves a derivative of the Christoffel symbol. So I think in a local frame $\nabla_aR_{bcde}$ is not $\partial_aR_{bcde}$.
Is there anything wrong?
