Consider the following definition of a root system from Brian C. Hall, Lie Groups, Lie Algebras, and Representations, GTM 222.
By 4. it follows that $\omega_\alpha(R)\subset R$. But later on, the author claims:
I agree that $\alpha = \omega_\alpha\cdot (\omega_\alpha \cdot \alpha)$ but I can not follow why this implies the onto statement.