Kernel of the Lie-algebra adjoint representation

Let $\mathfrak g$ be a Lie algebra. Its adjoint representation is

\mathrm{ad}:\mathfrak g\to \mathfrak{gl}(\mathfrak g),\qquad \mathrm{ad}_x(y)=[x,y].

Lemma.

\ker(\mathrm{ad}) \;=\; Z(\mathfrak g),

where $Z(\mathfrak g)$ is the center of $\mathfrak g$ .

Remarks

Proof. By definition, $x\in\ker(\operatorname{ad})$ if and only if $[x,y]=0$ for every $y\in\mathfrak g$ , which is precisely the condition $x\in Z(\mathfrak g)$ .

Thus $\operatorname{ad}_x$ depends only on the class of $x$ modulo the center.