I am studying the representation theory of some infinite dimensional algebras, for example infinite dimensional Lie-algebras, Kac-Moody algebras, W-algebras. These algebras arise as symmetry algebras of tow-dimensional conformally invariant quantum field theories.

I encountered the case that I have to study representations which are indecomposable, i.e. which cannot be decomposed into a direct sum of irreducible representations. However, in my cases, they always contain a irreducible sub-representation.

How can I name this irreducible sub-representation with a concise, short term so I don't have to write "the irreducible sub-representation of the indecomposable representation" all the time?

indecomposableisn't the one normally used in mathematics: instead, this word refers to a module which can't be written as the direct sum of two proper submodules (nothing being said about the irreducibility of submodules). Aside from this, the long quoted description in your header is probably the best you can do in general. Naturally you can always make up new words, but if you use established terms you need to accept their standard meanings. – Jim Humphreys Nov 14 '12 at 15:53