Let $G$ be an algebraic group over an algebraically closed field. Furthermore, let $G$ be semi-simple, i.e. its radical (viz. its maximal closed, connected, solvable normal subgroup) is trivial. One step in a proof I'm trying to understand seems to use the following fact:
If $G$ is connected and non-trivial, then there is a non trivial maximal torus in $G$.
Why is that true? Thank you!