Is it obvious that the Lie bracket of two horizontal field is a horizontal field?

So is it a trivial question if we ask the preimage of every hopf subalgebra is a coideal or cosubalgebra(we forget the antipod)?

@LSpice Thank you for your edit and your comment. But that comment was a complete and interesting answer to the main (first) part of the question. However the motivation part of that question is remained and may be need to be more discussed

@BenjaminSteinberg yes that is true for abelian group. The complete proof is given in the post for discrete abelian. In fact the surjectivity of the assembly map implies the Kaplanski conjecture. Anyway I am interested in the mitivation part of this post too.

I did updated the question. Thank you for your suggestion