I am trying to solve Exercsise 3(b) of Chapter 3 from Do Carmo's Riemannian Geometry. I am unable to understand a part of the hint, which I have framed as a question below.
Problem. Let $G$ be a Lie group equipped with a bi-invariant Riemannian metric (and endowed with the corresponding Riemannian connection). Then for any left invariant vector fields $X, Y$ and $Z$ in $G$, we have $\langle [X, Y], Z\rangle = \langle X, [Y, Z]\rangle$.
I am unable to make any progress.
I own Do Carmo's book as well, and the proposition in the hint is in fact proved on page 41 (chapter 1, section 2, in example 2.6 on Lie groups). I think the heart of the matter is that, as he shows, for left invariant vector fields, $$[Y,X] = \lim_{t\rightarrow 0}\frac{1}{t}\left(Ad(x_t^{-1}(e))Y - Y\right),$$ where $x_t$ is the flow of $X$ and $Ad$ the adjoint action of $G$ on its Lie algebra $\mathfrak{g}$. (This corresponds to another definition of the Lie bracket which I like more; roughly, defining it as the tangent map at the identity of the adjoint map from $G$ to $\mathfrak{g}$.) In any case, this allows one to write $$\langle U,V\rangle = \langle dR_{x_t(e)}U,dR_{x_t(e)}V\rangle$$ and differentiate with respect to $t$ to get the sought identity (up to some permutations using (anti-)symmetry).