= Solution
As a vector space, the <Lie algebra> of $G$ is $\mathfrak g=T_eG$. For $\xi\in\mathfrak g$, define
$$
l_\xi(g)=(dL_g)_e\xi,
$$
where $L_g$ is <left translation on a Lie group>. This vector field is smooth and left-invariant because $(dL_h)_g l_\xi(g)=l_\xi(hg)$. The assignment $\xi\mapsto l_\xi$ is linear and injective by evaluation at $e$. Conversely, every left-invariant vector field $V$ satisfies $V(g)=(dL_g)_eV(e)$, so it equals $l_{V(e)}$. Hence the map is an isomorphism.
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