The best I know of are some classification results for triangular Hopf algebras, which would be a subcase.  These are found in several papers by Etingof and/or Gelaki.  See [this paper](https://arxiv.org/pdf/math/0007154.pdf) and its references, for example.  Theorem 2.2.2.4 therein is a result of Kostant that generalizes your quoted result, I'll note.  I don't think even the triangular case has been completely determined, though I won't profess to be certain.

There are also some classification results on pointed quasitriangular Hopf algebras, such as [this paper on minimal such ones generated by skew primitives](http://www.ms.u-tokyo.ac.jp/journal/pdf/jms160404.pdf) by Masuoka.