TITLE: Is the sum of coroots the coroot of the sum?
QUESTION [7 upvotes]: 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.

REPLY [8 votes]: 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 $B$.