From killing equation to Lie algebra

Esmaeil
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Hi,
I want to know how can we arrive at the generators of Lie algebra if we have killing equation ?
on the other words, In this attached image I want to know how can I arrive at communication relation (16) by starting from killing vector (14) and its constraint (15)?https://www.physicsforums.com/images/attach/jpg.gif
 

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I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 
If you have the full set of killing vector fields, these vector fields (with constant coefficients) form a Lie algebra under the Lie bracket operation defined on the manifold...

You should be able to get the commutation relations just by taking lie brackets of the different killing vectors. Is this operation not working for you?
 
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