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

As a vector space, the Lie algebra of $G$ is $\mathfrak{g}=T_{e}G$. For $\xi \in \mathfrak{g}$, definewhere $L_g$ is left translation on a Lie group. This vector field is smooth and left-invariant because $(d{L}_h{)}_g{l}_{\xi }(g)=l_{\xi }(hg)$. The assignment $\xi \mapsto l_{\xi }$ is linear and injective by evaluation at $e$. Conversely, every left-invariant vector field $V$ satisfies $V(g)=(d{L}_g{)}_{e}V(e)$, so it equals $l_{V(e)}$. Hence the map is an isomorphism.