Root-system finiteness lemma

For roots $\alpha \ne \pm \beta$,where $\theta$ is their angle. Integrality gives an integer, and nonparallelism makes it strictly smaller than four.

Every reduced crystallographic rank-two root system is isomorphic to $A_1\times A_1$, $A_2$, $B_2$, or $G_2$. Their simple-root angles are respectively $\pi /2$, $2\pi /3$, $3\pi /4$, and $5\pi /6$.

The two simple reflections generate a dihedral group. For $A_1\times A_1$, $A_2$, $B_2$, and $G_2$, the rotation has order $2,3,4,6$, so the Weyl groups have orders $4,6,8,12$ respectively.