A very well-known theorem in Hopf algebra theory (see, for example, [this book][1] or the EGNO book) states that if $H$ is a cocommutative Hopf algebra over an algebraically closed field $\Bbbk$ of characteristic zero then $$H \cong U(P(H)) ~\#~ \Bbbk G(H)$$ 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][2] proved 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 Mrcun][3] proved an analogue of Milnor-Moore for bialgebroids and that, later, [Kalisnik and Mrcun][4] proved 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?


  [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