Let $G$ be a reductive group (over an algebraically closed field), $T$ a maximal torus, and $\Phi$ the root system of $(G,T)$. Then for each root $\alpha \in \Phi$ there is a unique connected $T$-invariant subgroup $U_\alpha$ (the root subgroup of $\alpha$) with Lie algebra $\mathfrak g_\alpha$. The Weyl group $W = N_G(T)/T$ acts simply transitively on $\Phi$ (by inner automorphisms), and one has $wU_\alpha w^{-1} = U_{w.\alpha}$ for $w \in W$ and $\alpha \in \Phi$.
Now let $G$ be a reductive $k$-group with $k$ a field which is not necessarily algebraically closed. Let $S$ be a maximal $k$-split $k$-torus of $G$ and $_k\Phi = {_k\Phi(G,S)}$ the relative root system, which may be non-reduced. The roots $\alpha \in {_k\Phi}$ with $\alpha/2 \notin {_k\Phi}$ form a reduced root system $_k\Phi_{nd}$, and for each $\alpha \in {_k\Phi}_{nd}$ one again has a unique connected unipotent subgroup $U_{(\alpha)}$ of $G$ normalised by $Z_G(S)$ with Lie algebra $\mathfrak g_\alpha$ or $\mathfrak g_\alpha \oplus \mathfrak g_{2\alpha}$ (the latter being the case if $2\alpha$ is a root). The relative Weyl group $_kW = N_G(S)(k)/Z_G(S)(k)$ again permutes the roots in $_k\Phi$.
My question: Does $wU_{(\alpha)}w^{-1} = U_{(w.\alpha)}$ hold in the relative case as well?
