The polynomiality of the Poisson center and semi-center of a Lie algebra and Dixmier's fourth problem

We show how Dixmier's fourth problem for an algebraic Lie algebra g can be reduced to that of its canonical truncation gΛ. Moreover, Dixmier's statement holds for all Lie algebras of dimension at most eight. The nonsolvable ones among them possess a polynomial Poisson center and semi-center.