Weyl group of a compact Lie group

Remarks

Example: for $G = SU(2)$ and $T = \set{\operatorname{diag}(z, \bar{z}) : \abs{z} = 1}$, the normaliser is $T \cup wT$ with $w = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Indeed $w \operatorname{diag}(z, \bar z) w^{-1} = \operatorname{diag}(\bar z, z)$, so $w \in N_G(T)$, and $W$ has order $2$, acting on $T$ by $z \mapsto \bar z$. Non-example: for the abelian group $G = \mathbb{T}^n$, the maximal torus is $G$ itself, $N_G(T) = T$, and $W$ is trivial; so $W$ does not detect anything in the abelian case. The group $W$ is finite, and it acts faithfully on $T$. Up to isomorphism it does not depend on $T$, since all maximal tori are conjugate. It is isomorphic to the Weyl group of the root system of $G$, the group generated by the reflections in the roots.