Fix a field $k$ of characteristic zero, and let $G$ be a connected reductive algebraic $k$-group of isotropic rank $\ge 1$. Fix a maximal $k$-split torus $S$, and let $\Phi_k$ be the relative root system of $G$ with respect to $S$. Assume that $\Phi_k$ is reduced and irreducible.

By Theorem 2 of Petrov and Stavrova, for every root $\alpha \in \Phi_k$ there is an embedding $X_\alpha:V_\alpha \to G$ where $V_\alpha$ is a vector group (EDIT: vector group scheme), and the image of $X_\alpha$ is the root subgroup $U_\alpha$. Given $v \in V_\alpha(k)$, $v \neq 0$, associated to $X_\alpha(v) \in U_\alpha(k)$ is a "Weyl group element" $w_\alpha(v)$, using Lemma 1.3 of Deodhar. If $G$ is a Chevalley group, these are the elements denoted $w_\alpha(t)$ in Steinberg, and $w_\alpha(1)$ is literally a representative of a Weyl group element. Steinberg lists some relations, of interest here is (R7), which says that for any two roots $\alpha,\beta \in \Phi_k$ and $v \in k$, $$ w_\alpha(1) \cdot X_\beta(v) \cdot w_\alpha(1)^{-1} = X_{w_\alpha \beta} \Big( c_{\alpha,\beta} v\Big) $$ This says that the action of the (lifts of) Weyl group elements on $G(k)$ by conjugation corresponds to the action of the Weyl group on $\Phi_k$; conjugating the root subgroup $U_\beta$ by a Weyl group representative $w_\alpha(1)$ takes it to the root subgroup $U_{w_\alpha \beta}$. The coefficient $c_{\alpha,\beta}$ is just a sign $\pm 1$ depending on $\alpha$ and $\beta$.

I have done a variety of explicit computations in quasi-split groups, and found in every case that $$ w_\alpha(u) \cdot X_\beta(v) \cdot w_\alpha(u)^{-1} = X_{w_\alpha \beta} \Big( c_{\alpha,\beta}(u,v) \Big) $$ for some function $c_{\alpha,\beta}:V_\alpha(k) \times V_\beta(k) \to V_{w_\alpha \beta}(k)$. Steinberg's (R7) says that in the split case $c_{\alpha,\beta}(1,v) = \pm v$. In the preprint (reference 3, equation 3) we give a version of this for a class of quasi-split special unitary groups, where $c_{\alpha,\beta}(1,v) = \pm v$ or $\pm \overline{v}$, with the bar denoting a Galois automorphism of order 2. I have also done computations in a quasi-split special orthogonal group, where some more complicated functions $c_{\alpha,\beta}$ arise.

**Question 1**: Why does the conjugation on the LHS above always end up in $U_{w_\alpha \beta}$, even in non-split cases? I suspect this is an obvious consequence of a Bruhat decomposition, but I don't understand that as well as I should.

**Question 2**: Is there a known generalization of this relation to reductive isotropic groups, or perhaps just for quasi-split groups?

*Deodhar, Vinay V.*,**On central extensions of rational points of algebraic groups**, Am. J. Math. 100, 303-386 (1978). ZBL0392.20027.2.*Petrov, V.; Stavrova, A.*,**Elementary subgroups of isotropic reductive groups.**, St. Petersbg. Math. J. 20, No. 4, 625-644 (2009); translation from Algebra Anal. 20, No. 4, 160-188 (2008). ZBL1206.20053.*Rapinchuk, I.; Ruiter, J.*, On abstract homomorphisms of some special unitary groups, arXiv:2107.07351, preprint.*Steinberg, Robert*,**Lectures on Chevalley groups**, University Lecture Series 66. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3105-1/pbk; 978-1-4704-3631-5/ebook). xi, 160 p. (2016). ZBL1361.20003.

notthe root group. So your worry is precisely about multipliable relative roots, right? $\endgroup$schemes, not ofgroups. (You didn't say otherwise, but I misread at first.) $\endgroup$7more comments