Quasi-Artinian Lie algebras (Q2266057)

Let L be a Lie algebra and \(L^{(r)}\) be the rth term in the derived series for L. L is called quasi-Artinian (Q.A.) if for every descending chain \(I_ 1\supseteq I_ 2\supseteq..\). of ideals of L, there exists an r such that \([L^{(r)},I_ r]\subseteq \cap_{i\geq 1}I_ i\). Q.A. contains all solvable algebras and all algebras which satisfy the minimum condition on ideals. The authors construct an example of an \(L\in Q.A\). which is neither solvable nor satisfies the minimum condition on ideals. A condition is found under which being Q.A. is equivalent to solvability and another under which Q.A. is equivalent to the minimum condition on ideals.