  Let M be a maximal subalgebra of the Lie algebra L. A subalgebra C of L is
said to be a completion for M if C is not contained in M but every proper
subalgebra of C that is an ideal of L is contained in M. The set I(M) of all
completions of M is called the index complex of M in L. We use this concept to
investigate the influence of the maximal subalgebras on the structure of a Lie
algebra, in particular finding new characterisations of solvable and
supersolvable Lie algebras.
