I am stuck in the following problem from Humphreys. Let $\alpha, \beta$ be roots in a root system $\Phi$. Let the $\alpha$-string through $\beta$ be $\beta - r\alpha, \ldots, \beta + q\alpha$ and let ...
$\Delta \subset \Phi$ 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 ...