For an algebraically closed field $k$, I'd like to show that the algebraic group $G=SL(n,k)$ is semisimple. Since $G$ is connected and nontrivial, this amounts to showing that the radical of $G$, denoted $R(G)$, is trivial. $R(G)$ can be defined as the unique largest normal, solvable, connected subgroup of $G$.

I know that the group of $n$th roots of unity of $k$ is inside of $G$, and it is normal and solvable (being in the center of $G$) but not connected, having one irreducible component for each root of unity. What are other normal subgroups in $G$? How can I show that $R(G)=e?$