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}.