I've been trying to proof this:
In a compact lie group with bi-invariant metric $g$, the sectional curvature holds the next equality.
$K(\sigma)=\frac{1}{4}\|[X,Y]\|^2$
When I tried, I needed to use that $g([[X,Y],X],Y)=g([X,Y],[X,Y])$ but I couldn't prove it.
Thank you
Best Answer
A left invariant metric on $G$ is given by a metric on ${\frak g}=T_e(G)$ that is extended to the tangent space $T_x(G)$ at point $x$ by left invariance. The metric thus obtained is bi-invariant if and only if the metric $g$ on $\frak{g}$ is invariant under the automorphisms $Ad(x)$ of $\frak{g}$, for all $x\in G$. This implies ( and is equivalent to if $G$ connected) that $g$ is invariant under the maps $ad(X)\colon Y \mapsto [Y,X]$, that is $$g([Z,X], Y) + g(Z, [Y,X])=0$$ for all $X$, $Y$, $Z$ in $\frak{g}$
Apply the above equality for $Z =[X,Y]$.