Conjugation action of a Lie group on itself

Let $G$ be a Lie group . The conjugation action of $G$ on itself is the map

This is a smooth left action: one has $C(e,h)=h$ and $C(g_1,C(g_2,h))=C(g_1g_2,h)$ for all $g_1,g_2,h\in G$. For each fixed $g$, the map $h\mapsto ghg^{-1}$ is a diffeomorphism of $G$, with inverse given by conjugation by $g^{-1}$.

Differentiating the conjugation map at the identity in the second variable yields the adjoint action on the Lie algebra: $\mathrm{Ad}_g=(\mathrm{d}C_g)_e$ where $C_g(h)=ghg^{-1}$.

Examples

1. Abelian groups. If $G$ is abelian, then $ghg^{-1}=h$ for all $g,h$, so the conjugation action is trivial and every conjugacy class is a point.
2. Similarity of matrices. For $G=\mathrm{GL}(n,\mathbb{R})$, conjugation is $C(A,B)=ABA^{-1}$; its orbits are similarity classes, which control Jordan normal form over algebraically closed fields.
3. Normal subgroups via conjugation. A subgroup $N\subset G$ is normal if and only if it is invariant under conjugation: $gNg^{-1}=N$ for all $g\in G$.