Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 4 / i / Solution

The crystallographic axiom makesintegers. If $\theta$ is the angle between the roots, thenThis is a nonnegative integer. Since $\beta \ne \pm \alpha$, the roots are not parallel, so ${\cos }^2\theta <1$. Thereforewhich is the root-system finiteness lemma.