An integrality theorem of root systems

Let ΦΦ be an irreducible root system and ΔΔ be a base for ΦΦ; it is well known that any root in ΦΦ is an integral combination of the roots in ΔΔ. In comparison to this fact, we establish the following result: Any indecomposable subset TT of ΦΦ is contained in the ZZ-span of an indecomposable linearly independent subset of TT.