$\Delta \subset \Phi$ is a base in a root system imples $\Delta^\vee \subset \Phi^\vee$ is a base in a root system
(the notation here is compatible with J.E. Humphrey's "Introduction to Lie Algebras and Representation Theory") Let $\Phi \subset E$ be a root system. Let $\Delta \subset \Phi$ be a base. I already ...