A very well-known theorem in Hopf algebra theory (see, for example, [Lorenz - A tour of representation theory][1] or the EGNO book ([Etingof, Gelaki, Nikshych, and Ostrik - Tensor categories](http://www-math.mit.edu/~etingof/egnobookfinal.pdf))) states that if $H$ is a cocommutative Hopf algebra over an algebraically closed field $\Bbbk$ of characteristic zero then $$H \cong U(P(H)) \mathbin\# \Bbbk G(H)$$ as Hopf algebras, where $P(H)$ is the space of primitive elements of $H$ and $G(H)$ the group of group-like elements. As far as I know, this theorem is attributed differently to many people, but I believe the list in the title should be comprehensive of all of them. For example, I know that the Milnor–Moore part of the theorem is the one stating that an irreducible cocommutative Hopf algebra $H'$ over a field of characteristic zero satisfies $H' \cong U(P(H'))$.

I know that Nichols proved, in [The Kostant structure theorem for $K/k$ Hopf algebras](https://doi.org/10.1016/0021-8693(85)90052-3), an analogue of the full CGKMM theorem for Hopf algebroids of the form $(K,H)$, where $K$ is a field extension of $\Bbbk$, that Moerdijk and Mrčun proved, in [On the universal enveloping algebra of a Lie algebroid][3], an analogue of Milnor–Moore for bialgebroids and that, later, Kališnik and Mrčun proved, in [A Cartier–Gabriel–Kostant structure theorem for Hopf algebroids][4], an analogue of the full Cartier–Gabriel–Kostant–Milnor–Moore for Hopf algebroids over the algebra $\mathcal{C}^{\infty}_c(\mathcal{M})$ of smooth functions with compact support over a smooth real manifold $\mathcal{M}$.

Is anybody aware of some further developments/extensions of this result to other classes of Hopf algebras (for example, weak Hopf algebras, general Hopf algebroids, (co)quasi-Hopf algebras, etc.)?

Otherwise, is there anybody aware of counter-examples that suggest this could not be generalized further in some direction?

**Edit:** Just to add some further motivation to this quest. Let $G$ be a linear algebraic group over an algebraically closed field $\Bbbk$ of characteristic zero. The coordinate algebra $\mathscr{O}(G)$ is a commutative Hopf algebra. If we consider the subspace $\mathscr{O}(G)^\circ$ of $\mathsf{Hom}_\Bbbk\left(\mathscr{O}(G),\Bbbk\right)$ formed by all those linear functionals which vanish on a finite-codimensional ideal, this is a cocommutative Hopf algebra. By the CGKMM Theorem we know that
$$\mathscr{O}(G)^\circ \cong U\left(P\left(\mathscr{O}(G)^\circ\right)\right) ~\#~ \Bbbk G\left(\mathscr{O}(G)^\circ\right).$$
It happens that $P\left(\mathscr{O}(G)^\circ\right) \cong \mathfrak{g}$, the tangent Lie algebra to $G$ at the neutral element, and $G\left(\mathscr{O}(G)^\circ\right) \cong \mathsf{Alg}_\Bbbk\left(\mathscr{O}(G),\Bbbk\right) \cong G$, so that
$$\mathscr{O}(G)^\circ \cong U\left(\mathfrak{g}\right) ~\#~ \Bbbk G,$$
where on the right-hand side $G$ is considered as a discrete group and $U(\mathfrak{g}) \cong \mathsf{Dist}_G$, the hyperalgebra of distributions on $G$.

Is there anybody aware of some further results like this one for groupoids?

  [1]: https://books.google.be/books?id=PUNwDwAAQBAJ&pg=PA496&lpg=PA496&dq=%22cartier-gabriel-kostant%20theorem%22&source=bl&ots=XbCJjSY9IN&sig=ACfU3U0LidrfkLOEOMSOpEJb-EgBqvnUaw&hl=it&sa=X&ved=2ahUKEwiEvpj2xu_pAhUF-qQKHWS7ACYQ6AEwAnoECBcQAQ#v=onepage&q=%22cartier-gabriel-kostant%20theorem%22&f=false
  [2]: https://core.ac.uk/download/pdf/81961325.pdf
  [3]: https://www.jstor.org/stable/20764270?seq=1
  [4]: https://www.sciencedirect.com/science/article/pii/S0001870812003477