Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 1 / 6 / b / Solution

If $\beta =-\alpha$, then both Cartan integers are $-2$. This opposite-root case satisfies the printed inner-product hypothesis and must be included.

Otherwise reducedness makes the roots nonproportional. Writing the angle as $\theta$, strict Cauchy-Schwarz inequality givesBoth factors are negative integers, so their product is $1,2$ or $3$. The full list is thereforeThe nonproportional pairs correspond respectively to angles $120^{\circ }$, $135^{\circ }$ and $150^{\circ }$; reversing the order of the roots reverses the unequal pairs.