Adjoint action of a Lie group

Let $G$ be a Lie group with Lie algebra Lie algebra $\mathfrak{g}=T_eG$. For each $g\in G$, consider the diffeomorphism of $G$ given by the conjugation map

The adjoint action (often written $\mathrm{Ad}$) is the map

Equivalently, $\mathrm{Ad}_g$ is the unique linear map such that for every $X\in\mathfrak{g}$, the left-invariant vector field associated to $\mathrm{Ad}_g X$ is the pushforward of the left-invariant vector field associated to $X$ under $C_g$.

The map $\mathrm{Ad}$ is a Lie group homomorphism $G\to \mathrm{GL}(\mathfrak{g})$, called the adjoint representation of $G$ on $\mathfrak{g}$.

Examples

1. Matrix Lie groups. For $G\subset \mathrm{GL}(n,\mathbb{R})$, one has $\mathrm{Ad}_g(X)=gXg^{-1}$ for $X\in\mathfrak{g}$.
2. Abelian Lie groups. If $G$ is abelian, then $C_g=\mathrm{id}$ for all $g$, hence $\mathrm{Ad}_g=\mathrm{id}_{\mathfrak{g}}$.
3. Rotations. For $G=\mathrm{SO}(3)$, identifying $\mathfrak{so}(3)$ with $\mathbb{R}^3$ via the cross-product isomorphism, $\mathrm{Ad}_g$ corresponds to the usual rotation of vectors in $\mathbb{R}^3$ by $g$.