While studying the book Introduction to Lie Algebras and Representation Theory, by Humphreys, I've came across a problem that seems simple, but I just cannot figure out:
If $\alpha, \beta \in \Phi$, where $\Phi$ is the root system, then $(\alpha + \beta)^\vee = \alpha ^\vee + \beta^\vee$?
Studying the algebras I already know it feels true, but I wasn't able to show a proper proof.
Any help would be appreciated.
If this were true, a root system would be isomorphic to its dual, which need not be the case. Look for a counterexample in type $B$.