Discrete centre of Lie group

Question

Can the centre of a (compact, semisimple) Lie group $G$ be discrete? If yes, under which conditions on $G$ this happens? Any references would be appreciated.

Answer 1

If $G$ is a compact semisimple Lie group, then its center $Z_G$ is always discrete. This is because if $\mathfrak{g}$ is the Lie algebra of $G$, then $$\mathrm{Lie}(Z_G)=Z_{\mathfrak{g}},$$ the center of $\mathfrak{g}$. But $Z_{\mathfrak{g}}=\{0\}$ since $\mathfrak{g}$ is semisimple. (This is because $Z_{\mathfrak{g}}$ is an abelian ideal, so also a solvable ideal.)