Given a Lie group G, let $\alpha_i$ be one of its character elements. Then we can define a subtorus of maximal torus $T$ of codimension 1 by $T_{\alpha_{i}}=(ker\alpha_{i})^{\circ}$. Here we identify a weyl group element $s_{j}$ as an element in $N(T)/T$ corresponding to a different root from $\alpha_{i}$. Is it true that for any $t\in T_{\alpha_{i}}$, there exists a $t'\in T_{\alpha_{i}}$ such that $ts_{j}=s_{j}t'$?
This is true for the Lie group of type $A$. For the general case, it's not so obvious that this is true. If this is not true in general, are there some counterexamples?
I will appreciate any comments or answer.
To turn comments into an answer: This result is not true in general.
Namely, what you are asking is whether: $$s_j^{-1}T_{\alpha_i}s_j \stackrel{?}=T_{\alpha_i}$$ But the Weyl group action on the roots is defined via $$\alpha_i(s_j^{-1}ts_j)=s_j(\alpha_i)(t)$$ and consequently we have: $$s_j^{-1} T_{\alpha_i}s_j =T_{s_j(\alpha_i)}$$
Now I'm relatively sure that as soon as $s_j(\alpha_i) \neq \pm \alpha_i$, we have $T_{\alpha_i} \neq T_{s_j(\alpha_i)}$, and such a pair of $s_j$ and $\alpha_i$ occurs in every root system except those that consist (thanks @Moishe Kohan) only of a sum of copies of $A_1$.