  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 $|H|$. 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 $m$, $c$, $|H|$, and $|F|$ whose nilpotency class is bounded in terms
of only $|H|$ and $c$. Examples show that the condition of the kernel $F$ being
cyclic is essential.
