Kurosh theorem for certain Koszul Lie algebras

A. Kurosh's theorem for groups [4] provides the structure of any subgroup of a free product of groups and its proof relies on Bass-Serre theory of groups acting on trees. In the case of Lie algebras, such a general theory does not exist and the analogue of Kurosh theorem is false in general, as it was first noticed by A.I. Shirshov in [10]. However, we prove that, for a class of positively graded Lie algebras satisfying certain local properties in cohomology, such a structure theorem holds true for subalgebras generated by elements of degree 1. Such class consists of Koszul Lie algebras, in which all the subalgebras that are generated in degree 1 are Koszul.