Let $\Phi$ be a root system of euclidean space $E$.
Suppose that a subset \Phi'\subset \Phi satisfies \Phi'=-\Phi' and if \alpha,\beta\in\Phi' and $\alpha+\beta\in \Phi$, then \alpha+\beta\in \Phi'.
I want to show that \Phi' is a root system in span($\Phi'$).
Actually, I proved other axioms. The final one is to show the following:
\sigma_{\alpha}(\beta)=\beta-<\beta,\alpha>\alpha\in \Phi', where $<\beta,\alpha>=2\frac{(\beta,\alpha)}{(\alpha,\alpha)}$.
How can I show it?