Let $G$ be a compact connected Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, let $T\le G$ be a maximal torus with Lie algebra $\mathfrak t$, and let $\Phi\subset X^*(T)$ be the root system of the complexified Lie algebra $\mathfrak g_{\mathbb C}$ with respect to $T$. For a character $\lambda\in X^*(T)$, write the same symbol for its differential $d\lambda_e:\mathfrak t\to i\mathbb R$ and for its complex-linear extension $\mathfrak t_{\mathbb C}\to\mathbb C$. For each root $\alpha\in\Phi$, let $H_\alpha\in\mathfrak t_{\mathbb C}$ be the coroot element normalized by $\alpha(H_\alpha)=2$, and define $s_\alpha:\mathfrak t_{\mathbb C}^*\to\mathfrak t_{\mathbb C}^*$ by $s_\alpha(\beta)=\beta-\beta(H_\alpha)\alpha$. Then there exists $n_\alpha\in N_G(T)$ such that the usual normalizer action $(n_\alpha\cdot\lambda)(t)=\lambda(n_\alpha^{-1}tn_\alpha)$ on $X^*(T)$ is the restriction of $s_\alpha$. In particular, $s_\alpha\in W(G,T):=N_G(T)/T$ and $s_\alpha(\Phi)=\Phi$.