A smooth connected group has a unique affine-normal pseudo-abelian reduction

Let $N$ be the largest subgroup from [F2]. Its represented quotient is smooth and connected by [F2] and has no nontrivial smooth connected affine normal subgroup by the same result. By [F1], it is pseudo-abelian. This proves existence over every field and identifies the specified subgroup.

Suppose $N^{\mathrm{\prime }}$ is another smooth connected affine normal subgroup with pseudo-abelian quotient $Q^{\mathrm{\prime }}=G\mathrm{/}{N}^{\mathrm{\prime }}$. The largest-subgroup property gives $N^{\mathrm{\prime }}\subset N$. By [F3] the image of $N$ in $Q^{\mathrm{\prime }}$ is smooth, connected, affine and normal, so [F1] makes that image trivial. Thus the inclusion of $N$ into $G$ factors through the scheme kernel $N^{\mathrm{\prime }}$ of the quotient map, giving $N\subset N^{\mathrm{\prime }}$. The two inclusions prove equality as subgroup schemes and uniqueness. AC is inherited from [F2]–[F3].