We have two reduced root systems $\Psi \supsetneq \Phi$ in a real vector space $V$ such that $\Phi$ is the root system of a Lie group (the dual Lie algebra of whose maximal torus is $V$), but all we know about $\Psi$ is that the root reflections $s_\beta$ in elements $\beta \in \Psi$ preserve the weight lattice $P$ corresponding to $\Phi$.
A geometric representation theorist told me it is standard in this case that there exists some other Lie group whose root system is $\Psi$, but he didn't have a reference handy. Where can I find a citation for this result?
Update: I'm told in comments this is due to Serre.
Additional question: Given this setup, can I arrange it so that the group whose root system is $\Psi$ contains that whose root system is $\Phi$?