Let $G$ be a connected compact Lie group and $S$ a torus subgroup contained in the maximal torus $T$. Denote by $R_+$ the set of positive roots on $\mathfrak{t}$. Let $N_G(S)$ and $Z_G(S)$ be the normalizer and centralizer subgroups of $S$ respectively. It can be seen that $N_G(S)/Z_G(S)$ is a subgroup of the Weyl group $W:=N_G(T)/T$. Define $\mathcal{A}$ to be the set $\{\alpha\in R_+|\ \alpha|_{\mathfrak{s}}\neq 0,\ w\cdot\alpha\in R_+\text{ for all }w\in N_G(S)/Z_G(S)\}$ and $\mathcal{B}:=R_+\setminus\mathcal{A}$.
Question 1: Does $\mathcal{C}:=\{w\cdot\alpha\ |\ \alpha\in\mathcal{B},\ w\in N_G(S)/Z_G(S)\}$ form a root subsystem of $R$?
Question 2: If the answer to question 1 is yes, is the Weyl group for the above root subsystem given by $N_G(S)/Z_G(S)$?
I would appreciate any references if the answers are known in the literature.
Here is an example which answers affirmatively to both questions above. Let $G=SU(3)$ and $S=\left.\left\{\begin{pmatrix}z&&\\ &z^{-1}&\\ &&1\end{pmatrix}\right| |z|=1\right\}$. Then $N_G(S)/Z_G(S)\cong\mathbb{Z}/2$ where the generator swaps the first two diagonal entries of elements of $S$. $\mathcal{A}=\{L_1-L_3, L_2-L_3\}$ and $\mathcal{B}=\{L_1-L_2\}$. $\mathcal{C}:=\{(L_1-L_2), (L_2-L_1)\}$ is a root system of type $A_1$, and its Weyl group is $\mathbb{Z}/2$, being isomorphic to $N_G(S)/Z_G(S)$.