= 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 $3$. Also
$$
\frac{(r,r)}{(s,s)}=\frac{a}{b}.
$$
The absolute pairs $(|a|,|b|)$ are $(1,1)$, $(1,2),(2,1)$, or $(1,3),(3,1)$. Hence
$$
\boxed{\frac{\|r\|}{\|s\|}\in\{1,\sqrt2,1/\sqrt2,\sqrt3,1/\sqrt3\}.}
$$
More precisely, angles $\pi/3,2\pi/3$ require equal lengths; angles $\pi/4,3\pi/4$ require a long-to-short ratio $\sqrt2$; and angles $\pi/6,5\pi/6$ require a ratio $\sqrt3$. The exclusion of orthogonality matters: orthogonal irreducible components can be scaled independently.
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