Barsotti-Chevalley over a perfect field: unique smooth affine normal subgroup

Apply the first exact reduction in [F1] to obtain its largest smooth connected affine normal subgroup $N$ and smooth connected pseudo-abelian quotient $A$. The completeness theorem in [F1] makes $A$ proper because $k$ is perfect. Its connectedness is geometric by [F2], so [F3] identifies it as an abelian variety and proves commutativity and projectivity. The quotient projection and its kernel give the exact fppf sequence by [F2].

Conversely an abelian variety has no nontrivial smooth connected affine closed subgroup: any such subgroup is proper as a closed subscheme, geometrically integral by [F2], and a point by [F2]. Thus an abelian quotient is pseudo-abelian. If another smooth connected affine normal $N^{\mathrm{\prime }}$ has abelian quotient, the uniqueness clause of the first reduction in [F1] gives $N^{\mathrm{\prime }}=N$. Its largest-subgroup clause also gives the stated maximality. AC and DC are inherited from the exact reductions and projectivity suppliers; they are included in the statement.