Indecomposable Lie algebras with nontrivial Levi decomposition cannot have filiform radical

It is shown that a semidirect sum g = s−→⊕Rr of a semisimple Lie algebra s and a solvable Lie algebra r with respect to a representation of s which does not decompose into a direct sum of ideals cannot have a radical r associated to a filiform Lie algebra. This proves that this class of nilpotent Lie algebras has none interest for the structure theory of nonsolvable Lie algebras.

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