In the representation theory of Lie groups (say, over $\mathbb{R}$ or $\mathbb{C}$), one can show that a Lie algebra homomorphism between the Lie algebras of two algebraic groups $G$ and $H$ always results in a Lie group homomorphism if $G$ is simply connected.

It seems like one might be able to make this argument over a general field. In place of fundamental group, we could ask that the etale fundamental group of $G$ be trivial. Does this allow us to show that a homomorphism of Lie algebras results in a homomorphism of groups? Then one could prove that the semisimplicity of the algebra and of the group are equivalent using this argument.