Adjoint representation of a Lie algebra

Let $\mathfrak{g}$ be a Lie algebra with bracket $[\cdot,\cdot]$. The adjoint representation (often written $\mathrm{ad}$) is the linear map

The defining property is that $\mathrm{ad}$ is a Lie algebra homomorphism:

where the bracket on the right is the commutator of endomorphisms. This identity is equivalent to the Jacobi identity in $\mathfrak{g}$.

If $\mathfrak{g}$ arises as the Lie algebra of a Lie group $G$, then $\mathrm{ad}$ is the differential at the identity of the adjoint action $\mathrm{Ad}:G\to \mathrm{GL}(\mathfrak{g})$.

Examples

1. Abelian Lie algebra. If $\mathfrak{g}$ is abelian, then $[X,Y]=0$ for all $X,Y$, so $\mathrm{ad}_X=0$ for every $X$.
2. Matrices. For $\mathfrak{g}=\mathfrak{gl}(n,\mathbb{R})$, $\mathrm{ad}_A(B)=AB-BA$. Thus $\mathrm{ad}_A$ measures the failure of $A$ to commute with other matrices.
3. $\mathfrak{so}(3)$. Under the identification $\mathfrak{so}(3)\cong\mathbb{R}^3$ with bracket given by the cross product, $\mathrm{ad}_x(y)=x\times y$.