  This paper is a continued investigation of the structure of Lie algebras in
relation to their chief factors, using concepts that are analogous to
corresponding ones in group theory. The first section investigates the
structure of Lie algebras with a core-free maximal subalgebra. The results
obtained are then used in section two to consider the relationship of two chief
factors of L being L-connected, a weaker equivalence relation on the set of
chief factors than that of being isomorphic as L-modules. A strengthened form
of the Jordan-Holder Theorem in which Frattini chief factors correspond is
also established for every Lie algebra. The final section introduces the
concept of a crown, a notion introduced in group theory by Gaschutz, and shows
that it gives much information about the chief factors
