How to compute radical and unipotent radical of an algebraic group?

Question

By definition, an algebraic group $G$ is reductive if its unipotent radical is $\{e\}$. The radical of an algebraic group is the identity component of its maximal normal solvable subgroup. The unipotent radical of $G$ is the set of all unipotent elements in the radical of $G$. Is there a general method to compute radical and unipotent radical of $G$?

We know that $GL_n$ is reductive so the unipotent radical of $GL_n$ is $\{e\}$. In particular, how to compute the radical and unipotent radical of $GL_n$? Thank you very much.

Answer 1

For $GL_n$ the typical way I know is to apply Lie-Kolchin's theorem.

A consequence of that theorem is that a connected solvable closed subgroup is conjugate to a subgroup of upper triangular matrices. Then since the subgroup is also normal, it must actually be upper triangular, but by symmetry also lower triangular and thus it must consist of diagonal matrices.

Then one easily checks that any normal subgroup consisting of diagonal matrices must actually consist of scalar matrices (by conjugating with matrices of the form $I + E_{i,j}$ where $E_{i,j}$ is the matrix with a $1$ at the $(i,j)$ position and $0$ elsewhere), so in this case the radical is just the center, and thus the unipotent radical is trivial.