Let $G$ be a Lie group with bi-invariant metric $\langle , \rangle$ and $X,Y,Z$ left invariant vector fields in $G$, how to conclude that $X\langle Y,Z\rangle=0$?
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The function $$ G \ni g \mapsto \langle Y_g, Z_g \rangle $$ is constant since $$ \langle Y_g, Z_g\rangle = \langle dL_g Y_e, dL_g Z_e\rangle = \langle Y_e, Z_e\rangle. $$ So differentiating it by $X$ gives 0. Note we have used the facts that $Y$ and $Z$ are left-invariant but $X$ need not be.