Could anybody help me to solve this problem with roots system?
Be $\Phi$ an irreducible root system. $\Phi^{+}$ a choice of positives roots in $\Phi$. Prove that if $(\alpha,\beta)\ge0$ $\forall \beta \in \Phi^{+}$ then $\alpha$ is the highest root among the roots with the same lenght of $\alpha$.
Thanx! :)