Prove that Killing vector fields form Lie algebra.

Question

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

Answer 1

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 $L_X g = 0$.