Kernel of ad and the center

Let $\mathfrak g$ be a Lie algebra . The adjoint representation is the linear map

Lemma.

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

Proof.
By definition, $x\in\ker(\mathrm{ad})$ iff $\mathrm{ad}_x(y)=0$ for all $y\in\mathfrak g$, i.e. $[x,y]=0$ for all $y$. This is exactly the defining condition for $x\in Z(\mathfrak g)$.

Context.
This identifies the failure of $\mathrm{ad}$ to be injective with central directions and clarifies the meaning of inner derivations : $\mathrm{ad}_x$ depends only on the class of $x$ modulo the center. At the group level, a related statement is ker(Ad) equals the group center (connected case) .