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2 Clarification

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

For

Let $\mathbf u$ be the Lie algebrasalgebra 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 $\mathbf u=C_{\mathbf u}(\mathbf t)\oplus\sum \mathbf u_\alpha$ for $\alpha\in\Delta(\mathbf g,\mathbf t)$(roots of . Here $\mathbf g$ w.r.tu_\alpha=\mathbf u\cap\mathbf g_\alpha$or equivalently the set {$X\in\mathbf u\mid [H,X]=\alpha(H)X \forall H\in\mathbf t$}. Let now$\mathbf t$)t_{max}$ be a maximal torus containing $\mathbf t$, and $\Delta(\mathbf g,\mathbf t_{max})$ the corresponding root system (this is the "usual" root system).

If An element $T$ of $\mathbf t_{max}$ is called regular if $\alpha(T)\neq\beta(T)$ and $\alpha(T)\neq 0$ for all roots $\alpha,\beta\in\Delta(\mathbf \alpha\neq\beta\in\Delta(\mathbf g,\mathbf t_{max})$and some element .

If the torus $T\in\mathbf \mathbf t$ (such an contains a regular element is called regular), $T$, 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 $\mathbf u_\alpha\neq 0$ then $\mathbf u_\alpha=\mathbf g_\alpha$, and $\mathbf u$ is a sum of root spaces.

1

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

For the Lie algebras, you have $\mathbf u=C_{\mathbf u}(\mathbf t)\oplus\sum \mathbf u_\alpha$ for $\alpha\in\Delta(\mathbf g,\mathbf t)$ (roots of $\mathbf g$ w.r.t. $\mathbf t$).

If $\alpha(T)\neq\beta(T)$ and $\alpha(T)\neq 0$ for all roots $\alpha,\beta\in\Delta(\mathbf g,\mathbf t_{max})$ and some element $T\in\mathbf t$ (such an element is called regular), the roots w.r.t. $\mathbf t$ are in bijection with those w.r.t. $\mathbf t_{max}$, the root spaces are 1-dimensional and $\mathbf u$ is a sum of root spaces.