Principal derivations and codimension one ideals in contact and Frobenius Lie algebras

The aim of this work is twofold. First, we give an inductive procedure to construct a Frobenius (resp. contact) Lie algebra from a contact (resp. Frobenius) Lie algebra. Second, we prove that all Frobenius Lie algebras can be constructed in this way, i.e., every Frobenius Lie algebra can be constructed as an extension of a contact Lie algebra by adding a distinguished element called principal derivation. Hence, classification of Frobenius Lie algebras will follow from classification of contact Lie algebras and every contact Lie algebra which admits a principal derivation is isomorphic to a subalgebra of As an example, we classify all 4-dimensional Frobenius Lie algebra.