A new construction of polarizations of solvable lie algebras

Let G be a real solvable Lie algebra and G*, the vector space dual. For φεG*, a construction of a polarization at φ is described. The construction is based on the use of maximal elements of G with respect to a partial order that depends on φ. It is proved that the construction breaks down only when G fails to satisfy the criterion of Brezin and Dixmier.