I've read that there is a structure theorem for commutative algebraic groups over an algebraically closed field $K$ namely they are the direct product of a semisimple group and a unipotent group.
- Is there an analog for $k$ a non-algebraically-closed field (say perfect)? I'm interested in both the cases for characteristic 0 and p.
- Are there other structure theorems for commutative affine groups over $k$? (e.g involving semidirect products or just extensions)