  Let g be a finite dimensional Lie algebra over an algebraically closed field
k of characteristic zero. We collect some general results on the Poisson center
of S(g), including some simple criteria regarding its polynomiality, and also
on certain Poisson commutative subalgebras of S(g). These facts are then used
to finish our earlier work on this subject, i.e. to give an explicit
description for the Poisson center of all indecomposable, nilpotent Lie
algebras of dimension at most seven. Among other things, we also provide a
polynomial, maximal Poisson commutative subalgebra of S(g), enjoying additional
properties. As a by-product we show that a conjecture by Milovanov is valid in
this situation. These results easily carry over to the enveloping algebra U(g).
