Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $B$ be a Borel subgroup with maximal torus $T$ and unipotent radical $U$. Let $\Phi^+ = \Phi(B,T)$ and $\Delta$ the base of $\Phi = \Phi(G,T)$ corresponding to $\Phi^+$.


A subset $\Psi$ of $\Phi^+$ is called **closed** if whenever $\alpha, \beta \in \Psi$, and $\alpha + \beta$ is a root, we have $\alpha + \beta \in \Psi$. If $\Psi$ is closed, then the root subgroups $U\_{\alpha} : \alpha \in \Psi$ directly span a closed, connected subgroup $U\_{\Psi}$ of $U$ which is normalized by $T$ and whose Lie algebra is $\bigoplus\limits\_{\alpha \in \Psi} \mathfrak g\_{\alpha}$.


Furthermore, if $\alpha \in \Phi^+$, and $\Psi \subseteq \Phi^+$ is closed, and all roots of the form $a \alpha + b \Psi$ for $a, b \in \mathbb{Z}^+$ lie in $\Psi$, then the root subgroup $U\_{\alpha}$ normalizes $U\_{\Psi}$. These facts are proved in Chapter 14 of Borel, *Linear Algebraic Groups*.


In particular, let $\Psi = \Phi^+ - \Delta$. It is easy to see that $\Psi$ is closed and normalized by every root subgroup $U\_{\alpha} : \alpha \in \Phi^+$, and in particular, $U\_{\Phi^+ - \Delta}$ is a normal subgroup of $U$. Moreover, it is a consequence of 8.32 in Springer's *Linear Algebraic Groups* that for any $x \in U\_{\alpha}$ and $y \in U\_{\beta}$, the commmutator $xyx^{-1}y^{-1}$ lies in $U\_{\Phi^+ - \Delta}$. From here one can argue that $U\_{\Phi^+ - \Delta}$ contains the derived group of $U$ by producing a homomorphism $U \rightarrow \prod\limits\_{\alpha \in \Delta} \mathbf G\_a$ with kernel $U\_{\Phi^+ - \Delta}$.


My question is, is $U\_{\Phi^+ - \Delta}$ exactly the derived group of $U$? For $G = \textrm{GL}\_n$, this does seem to be the case.



Reference Request: Derived group of $\mathscr R_u(B)$
I think the answer is yes.


We can assume that $G$ is semisimple. As you have seen already, and which is is clear from the commutation relations, we have $[U, U] \leq \prod\_{\alpha \in \Phi^+ - \Delta} U\_\alpha$. I guess you could show equality by using the Chevalley commutator relations, although you need to be careful when $p$ is small (e.g. $G$ of type $B\_n$ and $p = 2$).


Here is an idea for a different way to see this. The containment $[U, U] \leq \prod\_{\alpha \in \Phi^+ - \Delta} U\_\alpha$ shows that $$\dim U / [U, U] \geq \dim U / \prod\_{\alpha \in \Phi^+ - \Delta} U\_\alpha = \operatorname{rank} G$$


So it would be enough to show that $\dim U/[U,U] \leq \operatorname{rank} G$.


For this, a result of Steinberg states that there exists a regular unipotent element $u \in G$, which has the property that $\dim C\_G(u) = \operatorname{rank} G$. One can show that $u$ is contained in a unique Borel subgroup $B$ of $G$, and that $C\_G(u) = C\_B(u) = C\_U(u)Z(G)$, where $U$ is the unipotent radical of $B$. Proofs can be found in reference (\*) below.


In any case, we have a unipotent element $u \in U$ such that $\dim C\_U(u) = \operatorname{rank} G$. The conjugacy class $\operatorname{Cl}\_U(u)$ of $u$ in $U$ is contained in the coset $u[U,U]$ (proof: $g^{-1}ug = u[u,g]$), so $$\dim \operatorname{Cl}\_U(u) = \dim U - \dim C\_U(u) \leq \dim [U,U]$$ 


giving $\dim U/[U,U] \leq \operatorname{rank} G$.


(\*) See Chapter 4, sections 4.1 - 4.5 in



> 
> Humphreys, James E.:
>  *Conjugacy classes in semisimple algebraic groups.*
>  Mathematical Surveys and Monographs, 43. American Mathematical Society, Providence, RI, 1995. 
> 
> 
> 


or: 



> 
> Theorem 3.1 in:
> 
> 
> Steinberg, Robert:
>  *Regular elements of semisimple algebraic groups.*
>  Inst. Hautes Études Sci. Publ. Math. No. 25 1965 49–80. 
> 
> 
> and Lemma 4.3 in:
> 
> 
> Springer, T. A.: *Some arithmetical results on semi-simple Lie algebras.*
>  Inst. Hautes Études Sci. Publ. Math. No. 30 1966 115–141. 
> 
> 
> 




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PS. For further terms of the derived series and the lower central series, you can take a look at the following paper, where they are described when $p$ is not too small for $G$. 



> 
> Azad, H.; Barry, M.; Seitz, G.:
>  *On the structure of parabolic subgroups.*
>  Comm. Algebra 18 (1990), no. 2, 551–562. 
> 
> 
> 


