Radical of an algebraic group

The radical of an algebraic group is the identity component of its maximal normal solvable subgroup. For example, the radical of the general linear group $$\operatorname{GL}_n(K)$$ (for a field K) is the subgroup consisting of scalar matrices, i.e. matrices $$(a_{ij})$$ with $$a_{11} = \dots = a_{nn}$$ and $$a_{ij}=0$$ for $$i \ne j$$.

An algebraic group is called semisimple if its radical is trivial, i.e., consists of the identity element only. The group $$\operatorname{SL}_n(K)$$ is semi-simple, for example.

The subgroup of unipotent elements in the radical is called the unipotent radical, it serves to define reductive groups.