  A finite dimensional Lie algebra L with magic number c(L) is said to satisfy
Rentschler's property if it admits an abelian Lie subalgebra H of dimension at
least c(L) - 1. We study the occurrence of this new property in various Lie
algebras, such as nonsolvable, solvable, nilpotent, metabelian and filiform Lie
algebras. Under some mild condition H gives rise to a complete Poisson
commutative subalgebra of the symmetric algebra S(L). Using this, we show that
Milovanov's conjecture holds for the filiform Lie algebras of type Ln, Qn, Rn,
Wn and also for all filiform Lie algebras of dimension at most eight. For the
latter the Poisson center of these Lie algebras is determined.
