The article is available here, it seems that it hasn't been translated. The result you're interested in is Теорема 2; the proof is to notice that a non-abelian Lie algebra in characteristic zero contains as a subalgebra either a two-dimensional non-abelian Lie algebra or a three-dimensional Heisenberg Lie algebra, and their universal envelopings are not PI.

The article is available here, it seems that it hasn't been translated. The result you're interested in is Теорема 2; the proof is to notice that a non-abelian Lie algebra in characteristic zero contains as a subalgebra either a two-dimensional non-abelian Lie algebra or a three-dimensional Heisenberg Lie algebra, and their universal envelopings are not PI.

UPD: This only works for finite-dimensional Lie algebras.