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Abstract

This paper presents a systematic study of the structure of non-solvable cyclic metric Lie algebras. A cyclic metric is a symmetric bilinear form satisfying a cyclic cocycle condition, which arises naturally in the contexts of non-associative algebras and homogeneous pseudo-Riemannian manifolds. Firstly, we drive some sufficient conditions for a cyclic metric Lie algebra to be an orthogonal direct product of its semisimple and solvable parts. Then we introduce the notion of cyclic quadruples to analyze the interaction between semisimple and radical components. Finally, we use the double extension method to provide a complete characterization of non-degenerate cyclic metric Lie algebras that are neither semisimple nor solvable, over the fields of complex and real numbers.