Isomorphism question if base $\Delta$ for a root system is equal to the set of positive roots.

Question

Let $\Delta$ be a base for a roots system $\Phi$ in $E$. I'm trying to prove that if $\Delta = \Phi^{+}$ then $(\Phi, E)$ is isomorphic to an orthogonal direct sum of copies of $(\{\pm 1\}, \mathbb{R})$.

I wish I had a strategy, but I don't. If anyone could drop a hint as to how to prove this, I would be grateful.