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I want to find the integral curves of $[X,Y]$, then maybe can use this to prove. Can anyone gives an answer ? Thanks.

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Is your question the one in the title of this post? Or is it something else? – Willie Wong Mar 13 '12 at 15:17
Yes,just the question in the title. – henry Mar 14 '12 at 1:45
Just prove if $X$,$Y$ are both killing vector fields,then $[X,Y]$ is also killing vector field.Can anyone give some suggestions?Thanks. – henry Mar 14 '12 at 3:13

From the Jacobi identity and definition of Lie derivative it follows that $L_{[X,Y]} = L_X \circ L_Y - L_Y \circ L_X$. Thus $L_{[X,Y]} g = 0$ if $L_Y g = 0$ and $L_X g = 0$.

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