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 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 by Masuoka.
Update (5/2/17): For some reason, an update to an old (1999) preprint of Etingof and Gelaki has been posted to the arxiv: The Classification of Triangular Semisimple and Cosemisimple Hopf Algebras Over an Algebraically Closed Field. Not sure if that was some automatic system recompiling of the article or legit update at this point. Either way, it's a paper worth looking at for the triangular case.