Nilpotency of Lie type algebras with metacyclic frobenius groups of automorphisms

Assume that a Lie type algebra admits a Frobenius group of automorphisms with cyclic kernel $F$ of order $n$ and complement $H$ of order $q$ such that the fixed-point subalgebra with respect to $F$ is trivial and the fixed-point subalgebra with respect to $H$ is nilpotent of class $c$. If the ground field contains a primitive $n$th root of unity, then the algebra is nilpotent and the nilpotency class is bounded in terms of $q$ and $c$. The result extends the well-known theorem of Khukhro, Makarenko, and Shumyatsky on Lie algebras with metacyclic Frobenius group of automorphisms.