Projection of root systems and the generalized injectivity conjecture for exceptional groups

Let a be a real euclidean vector space of finite dimension and Σ a root system in a with a basis Δ. Let Θ ⊂ Δ and M = MΘ be a standard Levi of a reductive group G such that aΘ = aM / aG. Let us denote d the dimension of aΘ, i.e the cardinal of Δ - Θ and ΣΘ the set of all non-trivial projections of roots in Σ. We obtain conditions on Θ such that ΣΘ contains a root system of rank d. When considering the case of Σ of type exceptional, we give a list of all exceptional root systems that can occur in Σ_Θ and use it to prove the generalized injectivity conjecture in most exceptional groups cases.