gr.group theory - Unipotent linear algebraic groups

Assume characteristic 0. I do not know how much of this extends to finite characteristic.

Let $mathbf u$ be the Lie algebra of the unipotent subgroup $U$, and $mathbf t$ that of the torus (1- dimensional or not, it doesn't matter).

Define $Delta(mathbf g,mathbf t)$ as the sets of roots of $mathbf g$ w.r.t. $mathbf t$ (the usual definition is fine, even if $mathbf t$ is not maximal, however the root spaces will in general not be 1-dimensional). Let $C$ denote the centralizer.

Then you have $mathbf u=C_{mathbf u}(mathbf t)oplussum mathbf u_alpha$ for $alphainDelta(mathbf g,mathbf t)$. Here $mathbf u_alpha=mathbf ucapmathbf g_alpha$ or equivalently the set {$Xinmathbf umid [H,X]=alpha(H)X forall Hinmathbf t$}.

Let now $mathbf t_{max}$ be a maximal torus containing $mathbf t$, and $Delta(mathbf g,mathbf t_{max})$ the corresponding root system (this is the "usual" root system).
An element $T$ of $mathbf t_{max}$ is called regular if
$alpha(T)neqbeta(T)$ and $alpha(T)neq 0$ for all roots $alphaneqbetainDelta(mathbf g,mathbf t_{max})$.

If the torus $mathbf t$ contains a regular element $T$, the roots w.r.t. $mathbf t$ are in bijection with those w.r.t. $mathbf t_{max}$, and in particular the root spaces are 1-dimensional. It follows that if $mathbf u_alphaneq 0$ then $mathbf u_alpha=mathbf g_alpha$, and $mathbf u$ is a sum of root spaces.