Let $G$ be a compact connected Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, and let $T\le G$ be a maximal torus, meaning a compact connected abelian Lie subgroup maximal among compact connected abelian Lie subgroups of $G$, with Lie algebra $\mathfrak t$. Let $S^1:=\{z\in\mathbb C:|z|=1\}$. Let $\mathfrak g_{\mathbb C}:=\mathfrak g\otimes_{\mathbb R}\mathbb C$ and $\mathfrak t_{\mathbb C}:=\mathfrak t\otimes_{\mathbb R}\mathbb C$. For each continuous character $\lambda\in X^*(T):=\operatorname{Hom}_{\mathrm{cts}}(T,S^1)$, define the corresponding weight space

\begin{align*} \mathfrak g_\lambda:=\{X\in\mathfrak g_{\mathbb C}:\operatorname{Ad}(t)X=\lambda(t)X\text{ for every }t\in T\}. \end{align*}

Let

\begin{align*} R:=\{\lambda\in X^*(T):\lambda\ne 1\text{ and }\mathfrak g_\lambda\ne 0\} \end{align*}

be the set of non-zero roots as characters. For $\lambda\in R$, define the associated real root

\begin{align*} \alpha_\lambda:=-i\,d\lambda_e\in\mathfrak t^*, \end{align*}

where $e\in T$ is the identity element and $d\lambda_e:\mathfrak t\to i\mathbb R$ is the differential at $e$. Set

\begin{align*} \Phi:=\{\alpha_\lambda:\lambda\in R\}\subset\mathfrak t^* \end{align*}

and let $E:=\operatorname{span}_{\mathbb R}(\Phi)$. Equip $E$ with the Euclidean [inner product](/page/Inner%20Product) induced from any $\operatorname{Ad}(G)$-invariant inner product on $\mathfrak g$ restricted to $\mathfrak t$ and then dualized to $\mathfrak t^*$. Then $\Phi$ is a reduced crystallographic root system in $E$. If $\Phi=\varnothing$, this means the empty crystallographic root system in the zero-dimensional real [vector space](/page/Vector%20Space) $E=\{0\}$.