Adjoint representation: discrete kernel iff discrete center

Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$, and let $\mathrm{Ad}$ denote the adjoint representation $\mathrm{Ad}:G\to \mathrm{Aut}(\mathfrak{g})$.

Theorem. If $G$ is connected, then

where $Z(G)$ is the center of $G$ . In particular, $\mathrm{Ad}$ has discrete kernel if and only if $Z(G)$ is discrete.

This is often packaged as: the adjoint action is “almost effective” precisely when the center is discrete; compare effective actions (which correspond to trivial kernel).

Context. The key input is that $\mathrm{Ad}(g)$ is the differential at the identity of the conjugation action ; for connected groups, acting trivially on $\mathfrak{g}$ forces $g$ to commute with a neighborhood of the identity, hence with all of $G$. A standard formulation appears as the kernel-of-Ad equals the center lemma .