Lie algebras admitting a metacyclic Frobenius group of automorphisms

Suppose that a Lie algebra $L$ admits a finite Frobenius group of automorphisms $FH$ with cyclic kernel $F$ and complement $H$ such that the characteristic of the ground field does not divide $\mathrm{\mid }H\mathrm{\mid }$. It is proved that if the subalgebra $C_L(F)$ of fixed points of the kernel has finite dimension $m$ and the subalgebra $C_L(H)$ of fixed points of the complement is nilpotent of class $c$, then $L$ has a nilpotent subalgebra of finite codimension bounded in terms of $m$, $c$, $\mathrm{\mid }H\mathrm{\mid }$, and $\mathrm{\mid }F\mathrm{\mid }$ whose nilpotency class is bounded in terms of only $\mathrm{\mid }H\mathrm{\mid }$ and $c$. Examples show that the condition of the kernel $F$ being cyclic is essential.