Let $G$ be a finite-dimensional real Lie group with identity element $e$, and let $\mathfrak g=T_eG$ be its [Lie algebra](/page/Lie%20Algebra). Let $\exp:\mathfrak g\to G$ denote the Lie group exponential map, and let $\mathfrak X(G)=\Gamma(TG)$ denote the real [vector space](/page/Vector%20Space) of smooth vector fields on $G$. For each $a\in G$, let $L_a:G\to G$ denote left translation, $L_a(h)=ah$. For each $g\in G$, define the inner automorphism

\begin{align*} C_g:G\to G, \qquad h\mapsto ghg^{-1}, \end{align*}

and define

\begin{align*} \operatorname{Ad}_g:=d(C_g)_e:\mathfrak g\to \mathfrak g. \end{align*}

For $X\in\mathfrak g$, define the infinitesimal adjoint action

\begin{align*} \operatorname{ad}_X:\mathfrak g\to\mathfrak g \end{align*}

by

\begin{align*} \operatorname{ad}_X(Y)=\left.\frac{d}{dt}\right|_{t=0}\operatorname{Ad}_{\exp(tX)}Y. \end{align*}

If the Lie bracket on $\mathfrak g$ is defined by identifying each $X\in\mathfrak g$ with the unique left-invariant vector field $X^L\in\mathfrak X(G)$ satisfying $X^L_e=X$ and $X^L_a=d(L_a)_e(X)$ for all $a\in G$, and setting

\begin{align*} [X,Y]=[X^L,Y^L]_e, \end{align*}

then, for all $X,Y\in\mathfrak g$,

\begin{align*} \operatorname{ad}_X(Y)=[X,Y]. \end{align*}