This is a question in the context of Kottwitz' paper 'On the Hodge-Newton decomposition of split groups.'
Let $F$ be a finite extension of $\mathbb{Q}_p$ with DVR $\mathcal{O}_F$ and let $G$ be a reductive group defined over $\mathcal{O}_F$. Further let $T$ be a split maximal torus and let $B_1$ and $B_2$ be two adjacent Borel subgroups containing $T$. So we have a root $\alpha$ which is positive for $B_1$ and negative for $B_2$. Denote by $U_1$ and $U_2$ the unipotent radicals of the above Borels and by $U_\beta$ the root subgroup corresponding to any root $\beta$.
Then the claim ist that $U_1=(U_1\cap U_2)\rtimes U_\alpha$.
It is clear to me that $(U_1\cap U_2)$ and $U_\alpha$ generate $U_1$, as $U_1\cap U_2$ contains all root subgroups of $U_1$ except $U_\alpha$. But I don't see why $U_1\cap U_2$ is a normal subgroup of $U_1$. One of my ideas was to show that $U_\alpha$ normalizes every other root subgroup or at least permutes them. But I can't find any reference on the interaction of different root subgroups.
Do you have any ideas?