Assume that a Lie type algebra admits a Frobenius group
of automorphisms with cyclic kernel \documentclass[12pt]{minimal}
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\begin{document}$ F $\end{document} of order \documentclass[12pt]{minimal}
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\begin{document}$ n $\end{document} and complement \documentclass[12pt]{minimal}
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\begin{document}$ H $\end{document} of order
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\begin{document}$ q $\end{document} such that the fixed-point subalgebra with respect to \documentclass[12pt]{minimal}
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\begin{document}$ F $\end{document} is trivial and
the fixed-point subalgebra with respect to \documentclass[12pt]{minimal}
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\begin{document}$ H $\end{document} is nilpotent of class \documentclass[12pt]{minimal}
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\begin{document}$ c $\end{document}. If
the ground field contains a primitive \documentclass[12pt]{minimal}
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\begin{document}$ n $\end{document}th root of unity, then
the algebra is nilpotent and the nilpotency class is bounded in terms of \documentclass[12pt]{minimal}
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\begin{document}$ q $\end{document}
and \documentclass[12pt]{minimal}
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\begin{document}$ c $\end{document}. The result extends the well-known theorem of Khukhro, Makarenko, and Shumyatsky
on Lie algebras with metacyclic Frobenius group of automorphisms.
