Let $G$ be a compact, connected Lie group and $S$ a torus in $G$ **not** assumed maximal. Then conjugation in $G$ induces a faithful representation of $N = N_G(S)/Z_G(S)$ in the Lie algebra $\mathfrak s$ of $S$.

In all the instances I know where the image of $N \to \mathrm{GL}(\mathfrak s)$ is a reflection group, it is because there is a closed, connected subgroup $K$ of $G$ such that $S$ is its maximal torus, and $N$ is isomorphic to the Weyl group of $K$.

For instance, it happens

If $S$ is a maximal torus,

If $S$ is a circle (

**edit**: this memory of mine is**untrue**; I had shown this only in a few cases, so the answer is negative),If $S$ is the maximal torus in some $K$ such that $H^*(G;\mathbb Q) \to H^*(K;\mathbb Q)$ is surjective,

If $S$ is the maximal torus in the fixed-point set of some finite-order Lie group automorphism of $G$.

Does it **always** happen? If $N$ is a reflection group, is it always a $W_K$?

A less concise version of this question has already appeared on MSE.