  We introduce the concept of a double automorphism of an A-graded Lie algebra
L. Roughly, this is an automorphism of L which also induces an automorphism of
the group A. It is clear that the set of all double automorphisms of L forms a
subgroup in Aut(L). In the present paper we prove several nilpotency criteria
for a graded Lie algebra admitting a finite group of double automorphisms. We
also give an application of our results to groups admitting a Frobenius group
of automorphisms.
