Contracted Bianchi identities

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Start with the Bianchi identity^[4]

${\displaystyle R_{abmn;\ell }+R_{ab\ell m;n}+R_{abn\ell ;m}=0.}$

Contract both sides of the above equation with a pair of metric tensors:

${\displaystyle g^{bn}g^{am}(R_{abmn;\ell }+R_{ab\ell m;n}+R_{abn\ell ;m})=0,}$

${\displaystyle g^{bn}(R^{m}{}_{bmn;\ell }-R^{m}{}_{bm\ell ;n}+R^{m}{}_{bn\ell ;m})=0,}$

${\displaystyle g^{bn}(R_{bn;\ell }-R_{b\ell ;n}-R_{b}{}^{m}{}_{n\ell ;m})=0,}$

${\displaystyle R^{n}{}_{n;\ell }-R^{n}{}_{\ell ;n}-R^{nm}{}_{n\ell ;m}=0.}$

The first term on the left contracts to yield a Ricci scalar, while the third term contracts to yield a mixed Ricci tensor,

${\displaystyle R_{;\ell }-R^{n}{}_{\ell ;n}-R^{m}{}_{\ell ;m}=0.}$

The last two terms are the same (changing dummy index n to m) and can be combined into a single term which shall be moved to the right,

${\displaystyle R_{;\ell }=2R^{m}{}_{\ell ;m},}$

which is the same as

${\displaystyle \nabla _{m}R^{m}{}_{\ell }={1 \over 2}\nabla _{\ell }R.}$

Swapping the index labels l and m on the left side yields

${\displaystyle \nabla _{\ell }R^{\ell }{}_{m}={1 \over 2}\nabla _{m}R.}$