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...