Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 4 / 2 / ii / Solution

If the roots are parallel, their Euclidean norms are equal by reducedness. For nonparallel, nonorthogonal roots, the Cartan integers $a,b$ above are nonzero integers of the same sign and have product $1,2$ or $3$. AlsoThe absolute pairs $(|a|,|b|)$ are $(1,1)$, $(1,2),(2,1)$, or $(1,3),(3,1)$. HenceMore precisely, angles $\pi /3,2\pi /3$ require equal lengths; angles $\pi /4,3\pi /4$ require a long-to-short ratio $\sqrt{2}$; and angles $\pi /6,5\pi /6$ require a ratio $\sqrt{3}$. The exclusion of orthogonality matters: orthogonal irreducible components can be scaled independently.