There is a following theorem:

$H$ is a commutative Hopf algebra over a field $k$. Then there exists a bijective correspondence between $$\{ \textrm{Hopf subalgebras }K\subset H \} \quad \leftrightarrow\quad\{\textrm{ normal Hopf ideals } I\subset H\}$$

Where Hopf subalgebra is a subalgebra $K$ satisfying $\Delta K\subset K\otimes K$, and a normal Hopf ideal is a Hopf ideal $I$ which satisfies $ad(I)\subset I\otimes A$ where $ad$ is defined by $\mathrm{ad}:A\to A\otimes A$, $a\mapsto a_2\otimes a_1Sa_3$. The correspondence is defined by $$K\mapsto HK^+,\quad I\mapsto H^I:= k\square_{H/I}H=\{h\in H|\Delta h\equiv h\otimes 1\pmod{H\otimes I}\}$$.

This is proven in M.Takeuchi, A CORRESPONDENCE BETWEEN HOPF IDEALS AND SUB-HOPF ALGEBRAS (1972).

The following is my personal impression on the proof. The easy part is to show that $K\mapsto HK^+\mapsto H^{HK^+}$ is $K$, which depends on rather complicated proof of the faithful flatness of $H$ over $K$. For the other way around, the paper uses a variant of argument Chevalley's theorem on algebraic groups, which is not truely "algebraic".

I found some papers which simplifies the proof that $H$ is faithfully flat over $K$. Some of those are: Schneider, Principal Homogeneous Spaces for Arbitrary Hopf Algebras(1990) Masuoka and Wigner, Faithful Flatness of Hopf Algebras (1992)

However, I couldn't find any simplification of the proof of $H(H^I)^+=I$.

I would like to know about more recent result related to this theorem. Is there any simplification on the part that I mentioned? Is there any generalization for, like quantum groups?

I come to this question while I was learning basic theory of algebraic groups or group schemes. Most references construct a quotient by realizing it as an orbit space of certain action on a projective plane. Then I thought there could be a way of constructing it in terms of coordinate rings. I heard that there is a chapter in SGA which deals with a construction of quotient, but I don't have access to french literatures yet... I would appreciate if someone instead explain their approach a bit, or give some references written in english.

Thank you.