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 $B$.


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