If $G$ is a Lie group with a bivariant metric and if $U,V,X$ are left invariant vector fields, I wish to prove that $\langle[U,X],V\rangle=-\langle U,[V,X]\rangle$.
Following the proof of Do Carmo’s Riemannian Geometry book, I was able to understand the proof as far as the fact that $\langle U,V\rangle=\langle dx_tU,dx_tV\rangle$.
Now DoCarmo says differentiating with respect to $t$ at $t=0$ gives the result using the fact that $$[Y,X]=\lim_{t\rightarrow 0}\frac{dx_tY-Y}t.$$ But I do not know how to do this last step. How should I proceed?
Subtracting $\langle U, dx_t V\rangle$ from $\langle U,V\rangle=\langle dx_tU,dx_tV\rangle$ gives $$ \langle U,V-dx_t V\rangle=\langle dx_tU-U,dx_tV\rangle. $$ Hence, taking the limit when $t\to0$ yields the identity $$ -\langle U,[V,X]\rangle=\langle [U,X],V\rangle. $$