Products of smooth connected affine normal subgroups are in the same class

Conjugation of $N_2$ on the normal subgroup $N_1$ defines a semidirect product with underlying scheme $N_1\times N_2$ and multiplication $(n_1,n_2)(m_1,m_2)=(n_1(n_2{m}_1{n}_2^{-1}),n_2{m}_2)$. Multiplication and inverse are regular by the subgroup and group identities. The morphism to $G$ sending $(n_1,n_2)$ to $n_1{n}_2$ is a homomorphism. Its source is affine and smooth by products, and connected by [F2]. Thus [F1] gives a closed smooth connected affine image $P$. Since the exact image projection is fppf onto, the points of $P$ are precisely fppf local products, so $P=N_1{N}_2$. The identity in either factor gives inclusion of both subgroups.

Any point of $G$ conjugates each $N_i$ into itself after every base change. Conjugating a local product therefore gives another local product. Since membership in a closed subgroup is detected after a faithful flat cover by its ideal equations, $P$ is normal scheme theoretically. This proves every asserted property. AC is inherited from [F1]–[F2].