Let $G$ be a connected reductive group over a field $k$, $A_0$ a maximal $k$-split torus, $\Phi = \Phi(A_0,G)$, and $\Delta$ a base of $\Phi$. Let $P_0$ be a minimal $k$-parabolic subgroup containing $A_0$ with Levi subgroup $M_0 = Z_G(A_0)$, and let $N_0 = \mathscr R_u(P_0)$ with $\Phi(A_0,N_0) = \Phi^+$.
Let $\theta \subseteq \Delta$, let $w$ be in the relative Weyl group $N_G(A_0)/M_0$, and assume that $w (\theta) \subseteq \Delta$. For $S \subseteq \Delta$, let $P_S = M_S N_S$ be the corresponding standard parabolic subgroup, and let $[S] = \Phi(A_0,M_S)^+$.
I have read that $N_{w \theta} \cap wN_{\theta}w^{-1}$ is normal in $N_{w \theta}$, and I'm trying to understand why.
My approach was to first try the special case when $G$ is split over $k$. In this case, I can argue that $N_{w \theta} \cap wN_{\theta}w^{-1}$ is directly spanned in any order by the root subgroups $U_{\alpha} : \alpha \in E$, where
$$E = \{ \alpha \in \Phi^+ - [w\theta] : w^{-1} \alpha > 0 \} $$
Note that $\Phi(A_0,N_{w\theta}) = \Phi^+ - [w\theta] $.
In order to show that $N_{w\theta}$ normalizes $N_{w \theta} \cap wN_{\theta}w^{-1}$, I was trying to apply Proposition 14.5(3) in Borel, Linear Algebraic Groups. It comes down to showing this (replacing $w \theta$ by $S$, and $w^{-1}$ by $w$:
Let $S \subseteq \Delta$, assume $w(S) \subseteq \Delta$, and let $\alpha, \beta \in \Phi^+ - [S]$. Assume that $\alpha + \beta$ is a root. If $w(\beta) > 0$, and $w(\alpha) < 0$, then $w(\alpha + \beta) > 0$.
I did not have success with this approach. Is there a more sensible way to show that $N_{w \theta} \cap wN_{\theta}w^{-1}$ is normal in $N_{w \theta}$?
