Let K be an algebraically closed field of characteristic zero, g be a countably dimensional locally finite Lie algebra over K, and h is contained in g be a (a priori non-abelian) locally nilpotent subalgebra of g which coincides with its zero Fitting component. We classify all such pairs (g, h) under the assumptions that the locally solvable radical of g equals zero and that g admits a root decomposition with respect to h. More precisely, we prove that g is the union of reductive subalgebras g_n such that the intersections g_n ∩ h are nested Cartan subalgebras of g_n with compatible root decompositions. This implies that g is root-reductive and that h is abelian. Root-reductive locally finite Lie algebras are classified in [6]. The result of the present note is a more general version of the main classification theorem in [9] and is at the same time a new criterion for a locally finite Lie algebra to be root-reductive. Finally we give an explicit example of an abelian selfnormalizing subalgebra h of g = sl(∞) with respect to which g does not admit a root decomposition.
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