In this paper, we study simple ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}-Lie algebras and 4-dimensional ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}-Lie algebras over the field of complex numbers. We provide an approach to classify all 4-dimensional non-Lie complex ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}-Lie algebras. We prove that any non-Lie finite-dimensional simple ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}-Lie algebra has dimension 3. A complete list of all non-Lie complex simple ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}-Lie algebras is also derived.
