Let $G$ be a reductive group defined over a field $F$. Let $\Sigma$ be the set of roots of $G$ with respect to a Borel subgroup $B=TU$ with torus $T$. Let $W=N_G(T)/T$ be the Weyl group of $G$. For $\alpha\in \Sigma$, let $U_\alpha$ be the root space of $\alpha$. Denote $x_\alpha:F\rightarrow U_\alpha$ the fixed isomorphism.

Let $\alpha\in \Sigma$ be a root, and $w\in W$ be a Weyl element such that $w(\alpha)=\alpha$. My question is: is it true that $w$ has a representative $\dot w\in G$ such that $$\dot w x_\alpha(r)\dot w^{-1}=x_\alpha(r),\forall r\in F?$$

If this is false in general, in what cases it is true?

Thanks in advance.