Double automorphisms of graded Lie algebras

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. One of the obtained results is as follows. Let $A$ be a torsion-free abelian group and $L$ an $A$-graded Lie algebra in which $[L,\underbrace{L_0,ldots,L_0}_{k}]=0$. Assume that $L$ admits a finite group of double automorphisms $H$ such that $C_A(h)=0$ for all nontrivial $hin H$ and $C_L(H)$ is nilpotent of class $c$. Then $L$ is nilpotent and the class of $L$ is bounded in terms of $|H|$, $k$ and $c$ only. We also give an application of our results to groups admitting a Frobenius group of automorphisms.