Pseudo-abelian varieties under separable algebraic extension

First let $K\mathrm{/}k$ be finite Galois and let $N$ be the subgroup supplied by [F1] for $G_K$. Every semilinear Galois automorphism of $G_K$ takes $N$ to a smooth connected affine normal subgroup, hence fixes it by maximality. This stable closed subscheme descends to a closed subscheme $N_0\subset G$. Here is the ideal descent explicitly: on an affine chart $U=\mathrm{Spec}B\subset G$, its ideal $I\subset B{\otimes }_{k}K$ is stable. Choose trace-dual bases $c_i,d_i$ by [F3]. Their embedding matrices have transposed product equal to the identity by trace duality, hence also inverse product equal to the identity; the row for the identity embedding then gives that ${\sum }_i{d}_i\sigma (c_i)$ is $1$ for $\sigma =1$ and $0$ otherwise; thus for $b\in I$, $b={\sum }_i{d}_i{\sum }_{\sigma \in \mathrm{Gal}(K\mathrm{/}k)}\sigma (c_{i}b)\mathrm{.}$ The inner sums are invariant elements of $I$, so $I=(I\cap B){\otimes }_{k}K$; invariants of $B{\otimes }_{k}K$ are $B$, by coefficientwise fixed-field equality. These ideals glue on chart overlaps by faithful flatness and define $N_0$. Multiplication, inverse, identity and conjugation factor through it since their defining ideal pullbacks become zero over $K$. By [F2], $N_0$ is affine and smooth. It is connected because a disconnection would pull back to one of $N$. Consequently pseudo-abelianness forces $N_0$ trivial and $N$ trivial. This proves the finite Galois case.

For arbitrary separable algebraic $K\mathrm{/}k$, suppose $G_K$ has a nontrivial smooth connected affine normal subgroup $H$. This subgroup, its group and conjugation factorizations, and its affine presentation descend to some finite separable $L\mathrm{/}k$ inside $K$: choose a finite affine cover of $G$, finitely many ideal generators defining $H$, their finitely many gluing and factorization equations, and an affine finite-presentation model for $H$ and its inverse chart maps. Every coefficient belongs to a finite subextension; enlarge $L$ to contain the finitely many coefficients. Smoothness also spreads after enlarging $L$: on a finite cover of $H$ use its smooth presentations with invertible Jacobian minors, and include their coefficients and the equations giving the cover. Alternatively geometric regularity descends by [F2] once the model is defined. The descended $H_L$ is connected and nontrivial since scalar extension to $K$ is surjective on spaces and faithfully detects an identity isomorphism. Embed $L$ in a finite Galois $M\mathrm{/}k$ by [F3]. Smoothness, affineness and normality persist, and connectedness persists by [F3]. Thus $H_M$ is nontrivial and contradicts the finite Galois case. Finally $G_K$ remains smooth by base change and connected by [F3]. AC enters through [F1]–[F3].