Projection of root systems

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 $M = M_{\Theta}$beastandardLeviofareductivegroup$G$suchthat$a_\Theta = a_M / a_G$\mathrm{.}Letusdenote$d$thedimensionof$a_\Theta$,i\mathrm{.}ethecardinalof$\Delta - \Theta$and$\Sigma_{\Theta}$ 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 $d$.