Let $U_1$ be a unipotent group inside some Chevalley group $G$. For now, think of $G$ as being $SL_n(K)$ where $K$ is a field; then we can take $U_1$ to be a bunch of strictly upper triangular matrics. Assume if you like that $K$ is algebraically closed.

Now suppose that $U_1$ is normalized by a non-trivial torus $T_1$. Are there any general statements that can be made about the structure of $U_1\ ?$

For instance: let us assume that $T_1$ is $1$-dimensional, as this is the limiting case. I suspect that the following is true: if $r(t)$ is not equal to $s(t)$ for all positive roots $r,s$, and all elements $t$ in $T_1$, then $U_1$ is a product of root subgroups.

I haven't written down a proof of this statement, but doodling suggests that it is true! Indeed I suspect it is true of $T_1$ contains ANY element $t$ satisfying the given condition. I would like a more general statement though: covering the case where $r(t)=s(t)$ for particular positive roots $r$ and $s$.

I should note that I tend to automatically ignore small characteristic cases! Funny things can happen in this situation...