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 Radᵤ (G) of G. We investigate some properties of Radᵤ (G), and study those G for which Radᵤ (G) is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine k-groups.