Let $G$ be a compact connected semisimple Lie group, let $T\le G$ be a maximal torus, and let $\mathfrak t$ be the [Lie algebra](/page/Lie%20Algebra) of $T$. Let $R\subset X^*(T)$ be the root system of the complexified Lie algebra $\mathfrak g_{\mathbb C}$ with respect to $T$. Let $Q$ denote the root lattice generated by $R$, and let $P$ denote the full weight lattice of the same root system, equivalently the character lattice of the maximal torus in the simply connected compact semisimple Lie group with Lie algebra $\mathfrak g$.

Under the standard identification of these lattices inside the common real Cartan dual, one has

\begin{align*} Q\subset X^*(T)\subset P. \end{align*}

Moreover, if $\widetilde G$ is the simply connected compact semisimple Lie group with Lie algebra $\mathfrak g$, if $\widetilde T\le \widetilde G$ is the maximal torus covering $T$, and if $C\le Z(\widetilde G)$ is finite, then the quotient $\widetilde G/C$ has torus character lattice

\begin{align*} L_C:=\{\lambda\in P:\lambda(c)=1\text{ for every }c\in C\}. \end{align*}

The assignment $C\mapsto L_C$ gives an inclusion-reversing correspondence between finite central subgroups $C\le Z(\widetilde G)$ and intermediate lattices

\begin{align*} Q\subset L\subset P. \end{align*}

In particular, the adjoint quotient corresponds to the root lattice $Q$.