Abstract
Let L be a finite-dimensional Lie algebra over a field k of characteristic zero and let U(L) be its enveloping algebra with quotient division ring D(L). Let P be a commutative Lie subalgebra of L. In [O2] the necessary and sufficient condition on P was given in order for D(P ) to be a maximal (commutative) subfield of D(L). In particular, this condition is satisfied if P is a commutative polarization (CP) with respect to any regular f ∈ L and the converse holds if L is ad-algebraic. The purpose of this paper is to study Lie algebras admitting these CP’s and to demonstrate their widespread occurrence. First we have the following characterisation if L is completely solvable: P is a CP of L if and only if there exists a descending chain of Lie subalgebras
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