A result on generating root systems by linearly independent subsets

Let Φ be a root system in a finite dimensional Euclidean space FF and S   be a subset of Φ. Let the smallest in the collection of all root systems in FF which contain S  —i.e., the intersection of all such root systems—be denoted by R(S)R(S). It can be easily shown that Φ has linearly independent subsets X   such that R(X)=ΦR(X)=Φ—e.g., for any base Δ of Φ, R(Δ)=ΦR(Δ)=Φ. We prove a result that generalizes the preceding fact: If Ψ is any subset of Φ, then there exists a linearly independent subset S of Ψ such that  R(S)⊇ΨR(S)⊇Ψ. In the process of deriving the above one, we find a sufficient condition for a root system to be isomorphic to one of the root systems in {An,Dn+3:n∈N}{An,Dn+3:n∈N} and obtain a simple proof of the following known result on exceptional root systems: Let  k,ℓk,ℓbe integers such that  6⩽k⩽ℓ⩽86⩽k⩽ℓ⩽8; if X is a subset of the exceptional root system  E8E8such that  R(X)R(X)is isomorphic to  EℓEℓ, then for some linearly independent subset Y of X,  R(Y)R(Y)is isomorphic to  EkEk.