Why is it that many sources define simple (or almost-simple) linear algebraic group $G/k$ to be a connected, semisimple linear algebraic group such that every proper connected normal subgroup is trivial? Doesn't semisimplicity follow from the last condition?
I am told by a friend that part of the issue may be that semisimplicity is defined over $\overline{k}$, whereas simplicity is defined over $k$. Would it then do to define matters as follows?
"A linear algebraic group $G/k$ is semisimple if $G$ has no connected, non-trivial, abelian normal algebraic subgroup. It is said to be simple if it is semisimple and connected and has no connected, non-trivial normal algebraic subgroup defined over $k$."
I've heard from other friends that there are issues involving non-reduced varieties (but I am using classical foundations...) or non-perfect fields. Does defining simple algebraic groups on top of semisimple algebraic groups take care of such issues, or at least the second one (non-perfect fields)?