  A Lie algebra L is known to be nilpotent if it admits a grading by (Zp, +)
with support X not containing 0. It is also known that the class of L can be
bounded by some explicit function of |X|. We generalise this and other
classical results to gradings of Lie algebras by arbitrary groups with
arithmetically-free support. We then apply these results to automorphisms of
groups satisfying an identity.
