Axioms of metabelian Lie Q-algebras and U-algebras

This is the third paper in the series of three, which are in the series of papers, the aim of which is to construct algebraic geometry over metabelian Lie algebras. We give the recursive set of universal formulas, axiomatizing universal class of all matabelian Lie U-algebras, and the recursive set of quasiidentities, axiomatizing quasivariety of all matabelian Lie Q-algebras. We have come to the characterization of finite generated objects from these universal classes. We show connections between such algebras and diophantine projective varieties over a field.