  Let U(L) be the enveloping algebra of a finite dimensional Lie algebra L over
a field k of characteristic zero, Z(U(L)) its center and Sz(U(L)) its
semicenter. A sufficient condition is given in order for Sz(U(L)) to be a
polynomial algebra over k. Surprisingly, this condition holds for many Lie
algebras, especially among those for which the radical is nilpotent, in which
case Sz(U(L))=Z(U(L)). In particular, it allows the explicit description of
Z(U(L)) for more than half of all complex, indecomposable nilpotent Lie
algebras of dimension at most 7.
