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

Let $a$ be a real euclidean vector space of finite dimension and $\mathrm{\Sigma }$ a root system in $a$ with a basis $\mathrm{\Delta }$. Let $\mathrm{\Theta }\subset \mathrm{\Delta }$ and $M=M_{\mathrm{\Theta }}$ be a standard Levi of a reductive group $G$ such that $a_{\mathrm{\Theta }}$ $=a_M\mathrm{/}{a}_G$. Let us denote $d$ the dimension of $a_{\mathrm{\Theta }}$, i.e the cardinal of $\mathrm{\Delta }-\mathrm{\Theta }$ and ${\mathrm{\Sigma }}_{\mathrm{\Theta }}$ the set of all non-trivial projections of roots in $\mathrm{\Sigma }$. We obtain conditions on $\mathrm{\Theta }$ such that ${\mathrm{\Sigma }}_{\mathrm{\Theta }}$ contains a root system of rank $d$. When considering the case of $\mathrm{\Sigma }$ of type exceptional, we give a list of all exceptional root systems that can occur in ${\mathrm{\Sigma }}_{\mathrm{\Theta }}$ and use it to prove the generalized injectivity conjecture in most exceptional groups cases.