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$\Delta \subset \Phi$ 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 ...
