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.