Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 4 / f / Solution

Take $u=v=l_{\xi }$ and $w=l_{\eta }$ in the identity from part (e). For a left-invariant metric, all three scalar products are constant and $[l_{\xi },l_{\xi }]=0$, soThe curve ${\gamma }_{\xi }$ has velocity $l_{\xi }$, hence is a geodesic exactly when ${\nabla }_{l_{\xi }}{l}_{\xi }=0$. Nondegeneracy of the inner product now givesThis is the geodesic-vector criterion for a left-invariant metric.