Working on an assignment I came across the following problem:
Show explicitly the Bianchi identity: $$ R^{a}_{b[cd;e]} = 0 $$ where ; denotes covariant differentiation $$ R^{a}_{bcd} = \frac{1}{2}g^{ae}(\partial_b \partial_cg_{ad}-\partial_d\partial_bg_{ac}-\partial_a\partial_cg_{ad}) $$$$ R^{a}_{bcd} = \frac{1}{2}g^{ae}(\partial_b \partial_cg_{ed}-\partial_d\partial_bg_{ec}+\partial_e\partial_d g_{bc}-\partial_e\partial_cg_{bd}) $$ However I am having trouble expanding the tensor since I do not know how to apply the notation to the LHS.
