For nilpotent connected groups, the semisimple elements form the unique maximal torus, and the uniqueness makes it Galois-stable, hence defined over \(F\). For a non-nilpotent group, one uses centralizers of noncentral semisimple elements in \(\operatorname{Lie}(G)(F)\). These centralizers are proper connected \(F\)-subgroups, so induction on dimension gives maximal \(F\)-tori inside them. The general case is reduced to the good case by an isogeny.

The generation assertion follows because the closed subgroup generated by the Cartan subgroups attached to maximal \(F\)-tori has full Lie algebra and is therefore all of \(G\). A semisimple \(F\)-point lies in the \(F\)-defined centralizer of itself, so the existence theorem applied to that centralizer puts it in a maximal \(F\)-torus.

For reductive \(G\), Cartan subgroups are maximal tori. Since \(F\)-tori are unirational and rational points on a unirational variety are dense over an infinite field, a finite product of maximal \(F\)-tori dominating \(G\) gives unirationality and density.