Let $G$ be a complex, linear algebraic group and $H\subseteq G$ a closed and normal subgroup. Then, the quotient $G/H$ has the structure of a affine variety. I am looking for the most "modern" reference in English language for this statement. I would prefer a graduate level textbook most. Anything I find, however, restricts to the reductive case, which is not general enough for me.
Given an affine (not necessarily algebraic) group $G$ over an arbitrary field and its closed normal subgroup $H$, the fpqc quotient sheaf $G/H$ is affine. A simple proof of this fact, which uses the language of Hopf algebras, was given by M. Takeuchi, A correspondence between Hopf ideals and sub-Hopf algebras. Manuscripta Math. 7 (1972), 251–270. The proof of the theorem in Section 16.3, of Waterhouse, Introduction to affine group schemes, Springer GTM 66, seems based on Takeuchi's idea.