An Integrality Theorem of Root Systems

Let R and Z denote the set of reals and the set of integers respectively and let E be a finite dimensional vector space over R with usual innerproduct (∗,∗). Let S and T be two subsets of E. If S is contained in the Z-span of T- or equivalently, if every vector of S is an integral combination of vectors in T- then we say that S is generated by T. et S of E is called decomposable if there is a proper subset T of S such that for all x ∈ T and for all y ∈ S \ T, (x, y) = 0; otherwise it is called indecomposable. be a 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. A natural question to ask in this connection is the following: If S is a linearly dependent subset of Φ, can there be a linearly independent subset of S which generates S? We answer this question affirmatively.