Let $G$ be a compact connected Lie group with identity element $e$ and [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 $G$. Let $\mathfrak t$ be the Lie algebra of $T$. Define
\begin{align*}
\mathfrak g_{\mathbb C}:=\mathfrak g\otimes_{\mathbb R}\mathbb C
\end{align*}
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and
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\begin{align*}
\mathfrak t_{\mathbb C}:=\mathfrak t\otimes_{\mathbb R}\mathbb C.
\end{align*}
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Let $R\subset X^*(T)$ be the finite set of non-trivial weights $\alpha$ for the complexified adjoint representation of $T$ on $\mathfrak g_{\mathbb C}$, so that the weight-zero space is $\mathfrak t_{\mathbb C}$ and