Fix a field $k$. This is perhaps unnecessary, but assume $\operatorname{char} k = 0$.
Let $G$ be a reductive isotropic quasi-split algebraic $k$-group. Let $S \subset G$ be a maximal split torus (of rank $\ge 1$ as $G$ is isotropic). Let $\mathfrak{g}$ be the Lie algebra, $\Phi_k$ be the relative root system of $G$ with respect to $S$, and for $\alpha \in \Phi_k$, let $\mathfrak{g}_\alpha$ be the associated root space. $$ \mathfrak{g}_\alpha(k) = \{ X \in \mathfrak{g}(k) : \operatorname{Ad}(s) X = sXs^{-1} = \alpha(s) X, \forall s \in S(k) \} $$ For a root $\alpha$, let $U^{\alpha} = \exp \mathfrak{g}_\alpha$ be the associated root subgroup, and let $W = N(S)/T(S)$ be the Weyl group of $G$. All of this I mostly understand.
My first problem is: I know that $W$ is isomorphic to the Weyl group of $\Phi_k$, but I don't know exactly how this isomorphism is expressed, and I cannot find a concise description of this identification anywhere.
My second problem is with the following claim: Let $\overline{w} \in W$, and choose a representative $w \in N(S)$, and let $x \in U^{\alpha}$. Then $$ wxw^{-1} \in U^{w(\alpha)} $$ (Technically $w$ is in the $k$-points of $N(S)$ and $x$ is in the $k$-points of $U^\alpha$, but this is not important right now.) This claim arises in remark 1.6 of Deodhar's thesis On central extensions of rational points of algebraic groups.
I don't understand why this claim is true. I suspect this is mostly due to my lack of understanding of exactly how $N(S)/Z(S)$ is identified with the Weyl group of the root system $\Phi_k$.
Additionally, it should be possible to say more about $wxw^{-1}$ than just that it lands in $U^{w(\alpha)}$. For example, in the split case $G = \operatorname{SL}_n$ Milnor gives explicit descriptions of such conjugations in corollary 9.4 of Introduction to Algebraic K-Theory. Is there any source which has more general versions of these relations for quasi-split groups of types other than $A_n$? Perhaps some of this is in Deodhar's thesis, but I have not yet found it.