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?