An integrality theorem of root systems

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