[Math] n almost-direct product decomposition for disconnected reductive algebraic groups

Question

$\textbf{Some definitions:}$
Let $G$ be an algebraic group (for me that is the complex points of an affine algebraic group). We say $G$ is reductive if its unipotent radical (maximal connected normal unipotent subgroup) $R_u(G)$ is trivial and we say $G$ is semisimple if its radical (maximal connected normal solvable subgroup) $R(G)$ is trivial. The derived group $\mathcal{D}(G)$ of $G$ is the intersection of the normal subgroups $N$ such that $G/N$ is commutative. If there are algebraic subgroups $H$ and $K$ of $G$ such that the product morphism $H\times K\rightarrow G$ is surjective with finite kernel we say that $G$ is an almost-direct product of $H$ and $K$. (edit: as pointed out in the comments/answers I forgot to specify that the images of $H$ and $K$ commute in $G$.)

$\textbf{A theorem:}$
When $G$ is a connected reductive complex algebraic group, it is the almost-direct product of a central torus $R(G)$ and a semisimple group $\mathcal{D}(G)$. (See page 181 in Borel's book, or page 168 in Humphreys' book "Linear Algebraic Groups".)

$\textbf{My question:}$
I would like to know to which extent this decomposition fails to generalize in the case where $G$ is disconnected. More precisely, I would be interested to know if there are similar decomposition results for a reductive group $G$ (not necessarily connected) that happens to be nilpotent or solvable.

Answer 1

A solvable reductive group has a very simple structure - the component group is solvable, and the connected component of the identity is an algebraic torus. Unfortunately it is not really possible to decompose it further, because the conjugation action would need to decompose as well, but there are many indecomposable actions of solvable finite groups on tori. These basically boil down to irreducible solvable subgroups of $GL_n(\mathbb Z)$, $n$ the dimension of the torus. There are lots of these.

For nilpotent groups, the situation is a bit better. The conjugation action must be trivial, so you can write the group as the almost direct product of a central torus and a finite group as follows: The torus is the connected component of the identity. The finite group is the kernel of the universal map from the group to a torus.