Proof of Killing Vectors Commutator Theorem

[Altabeh]

I was wondering if you could help me with the proof of the following theorem.

If $$A^{{\mu }}$$ and $$B^{{\mu }}$$ are Killing vectors, then so is their commutator $$C^{{\mu }}=[A,B]^{\mu}$$.

Thanks in advance

 

[diazona]

What have you tried?

 

[Altabeh]

What have you tried?

Assuming $$g_{\mu \nu}$$ as our metric, I can write

$$g_{\mu \alpha}A^{\alpha}_{;\nu}=-g_{\alpha\nu}A^{\alpha}_{;\mu}$$ and
$$g_{\mu \alpha}B^{\alpha}_{;\nu}=-g_{\alpha\nu}B^{\alpha}_{;\mu}$$.

And I can correlate these two to each other, but I'm afraid about the commutator. I just need a clue as to how the commutator is written in terms of $$A^{\mu}$$ and $$B^{\mu}$$.