Let $G$ be a real Lie group of dimension $n\in\mathbb N$ with identity element $e$, and let $\mathfrak g=T_eG$ be its tangent space at the identity, regarded as a real [vector space](/page/Vector%20Space). For each $a\in G$, let $L_a:G\to G$ denote the left translation map $L_a(g)=ag$. Let $\Gamma(TG)$ denote the real vector space of smooth sections of the tangent bundle $TG\to G$, and let $\mathfrak X_L(G)$ be the real vector space of left-invariant smooth vector fields on $G$, meaning

\begin{align*} \mathfrak X_L(G)=\{Y\in \Gamma(TG):(dL_a)_g(Y_g)=Y_{ag}\text{ for all }a,g\in G\}. \end{align*}

For each $X\in\mathfrak g$, define the set-theoretic section $\widetilde X:G\to TG$ by

\begin{align*} \widetilde X_g=(dL_g)_e(X). \end{align*}

Then $\widetilde X$ is a left-invariant smooth vector field, and the map $\Phi:\mathfrak g\to \mathfrak X_L(G)$ defined by $\Phi(X)=\widetilde X$ is an isomorphism of real vector spaces.