On Root_system: Elementary_consequences_of_the_root_system_axioms (wikipedia), from the relation $\langle \alpha, \beta \rangle = (2\cos(\theta))^2 \in \mathbb Z$, the value $\cos(\theta)$ can only be $0, \pm \frac{1}{2}, \pm \frac{\sqrt{2}}{2}, \pm \frac{\sqrt{3}}{2} \text { and } \pm \frac{\sqrt{4}}{2}= \pm 1$, corresponding to angles of 90°, 60° or 120°, 45° or 135°, 30° or 150°, and 0° or 180°. Then it says the following, which I did not follow
Condition 2 says that no scalar multiples of α other than 1 and -1 can be roots, so 0 or 180°, which would correspond to 2α or -2α, are out.
Why do $0$ or $180$ degrees between $\alpha$ and $\beta$ correspond to $2\alpha$ or $-2\alpha$, which obviously cannot exist in a root system containing $\alpha$? In other words, how does the angle between two roots $\alpha, \beta$ give information about the existence of the other root $\pm 2 \alpha$?
The reason seems to be that $\frac{\|\beta\|}{\|\alpha\|} = 2$. But I do not know if this is true, and if this is the reason why from this it follows that the angle between $\alpha$ and $\beta$ can not be 180°. Probably this has to do with $\langle \beta, \alpha \rangle = \frac{2(\beta,\alpha)}{(\alpha,\alpha)} = 2\frac{\|\beta\|}{\|\alpha\|}$?