  The Kurosh theorem for groups 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
exists and the Kurosh theorem is false in general, as it was first noticed by
Shirshov. 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 in degree 1. Such class consists of Lie
algebras, which have all the subalgebras generated in degree $1$ that are
Koszul.
