Given any affine algebraic group $G$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$ with a faithfull representation in $GL\_n(\mathbb{F})$ . If one knows the generators of the corresponding ideal, what can be said about the generators of $G^u$. Here $G^u$ shall denote the group generated by all unipotent elements of $G$. (Unlike the case where $G$ is irreducible and solvable, this group is not necessarily unipotent).


I am particular interested in bounds on the degrees of the generators; also any reference, which deals with unipotent generated groups is welcome.



generators of the ideal of an unipotent-generated algebraic group
Suppose we are over an algebraically closed field, and $G$ is connected. Then, we have an exact sequence 
$$
1\to U\to G\to G\_r\to 1, 
$$ 
where $U$ is the unipotent radical of $G$, and $G\_r$ is a reductive group. 
Since a semisimple or unipotent group is generated by unipotent elements, this implies that $G^u$ is the intersection of the kernels 
of all the characters of $G$. Characters of $G$ are grouplike elements of the Hopf algebra 
${\mathcal O}(G)$. So the additional relations are that some grouplike elements $g\_j\in {\mathcal O}(G)$ generating the group of characters of $G$ are equal to $1$. 


