Unipotent normal subgroups of algebraic groups

Let G be an affine algebraic group scheme over a field k . We show there exists a unipotent normal subgroup of G which contains all other such subgroups; we call it the restricted unipotent radical Rad u ( G ) of G . We investigate some properties of Rad u ( G ) , and study those G for which Rad u ( G ) is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine k -groups.