= 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> gives
$$
n_{\alpha,\beta}n_{\beta,\alpha}=4\cos^2\theta<4.
$$
Both factors are negative integers, so their product is $1,2$ or $3$. The full list is therefore
$$
\boxed{\begin{array}{c|c}
n_{\alpha,\beta}&\text{possible }n_{\beta,\alpha}\\\hline
-1&-1,-2,-3\\
-2&-1\quad\text{or }-2\text{ for opposite roots}\\
-3&-1
\end{array}}
$$
The nonproportional pairs correspond respectively to angles $120^\circ$, $135^\circ$ and $150^\circ$; reversing the order of the <roots> reverses the unequal pairs.
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