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Let M be a maximal subalgebra of the Lie algebra L. A subalgebra C of L issaid to be a completion for M if C is not contained in M but every propersubalgebra of C that is an ideal of L is contained in M. The set I(M) of allcompletions of M is called the index complex of M in L. We use this concept toinvestigate the influence of the maximal subalgebras on the structure of a Liealgebra, in particular finding new characterisations of solvable andsupersolvable Lie algebras.

The index complex of a maximal subalgebra of a Lie algebra