With help from Desmos |

\nabla_\nu\left[A,B\right]_\lambda+\nabla_\lambda\left[A,B\right]_\nu=0

$$The first problem was that I only knew (from way back) that the commutator can be expressed as $$

\left[A,B\right]^\lambda=A^\rho\partial_\rho B^\lambda-B^\rho\partial_\rho A^\lambda

$$which is not much use in this case. But it was easy to get to these two$$

\left[A,B\right]^\lambda=A^\rho\nabla_\rho B^\lambda-B^\rho\nabla_\rho A^\lambda

$$$$

\left[A,B\right]_\sigma=A^\rho\nabla_\rho B_\sigma-B^\rho\nabla_\rho A_\sigma

$$which are much better. I also needed a covariant form of the Riemann tensor which is$$

R_{\ \ \sigma\nu\mu}^\tau X_\tau=\left[\nabla_\mu,\nabla_\nu\right]X_\sigma

$$They all earned a place in Important Equations for General Relativity.

It was another 13 equations to get to the desired conclusion with evil side-tracks and errors on the way. I have not included those in the answer.

The answer does include much index juggling, including use of antisymmetries to really throw the Riemann tensor indices about. That was the final breakthrough. It's at

Commentary 3.8 Symmetries and Killing vectors.pdf (pages 4-6).

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