Arithmetically-free group-gradings of Lie algebras: II

We study Lie algebras L that are graded by an arbitrary group (G,⁎) and have finite support, X. We show that L is nilpotent of |X|-bounded class if X is arithmetically-free. Conversely: if a finite subset Y of G is not arithmetically-free, then Y supports the grading of a non-nilpotent Lie algebra.