Orbits of the group $\mathbf{GL}(r,k[X_1,\dots,X_n])$

In this paper it is shown that the study of projective metabelian Lie algebras of finite rank reduces to a partial solution of Serre's problem on projective modules over polynomial rings. It is also observed that projective commutative-associative algebras of dimension 1 are isomorphic to the ring of polynomials in one variable over the ground field.