Composition series of a module ($0=M_0\subset M_1\subset \cdots \subset M_r=M$)

A composition series of a module is a finite chain of submodules whose nonzero successive quotient modules are simple modules. Every finite-dimensional Lie algebra representation has such a chain, viewed as a module over the universal enveloping algebra.

A composition factor is an irreducible successive quotient in a composition series of a module. For a central Casimir operator, its scalar on a composition factor is an eigenvalue of the operator on the original Lie algebra representation.