= Solution
Choose simple roots $\alpha,\beta$. Their inner product is nonpositive, so their angle lies in $[\pi/2,\pi)$. Part i leaves four possibilities for $mn$:
* $mn=0$: angle $\pi/2$, giving $A_1\times A_1$.
* $mn=1$: angle $2\pi/3$ and equal root lengths, giving $A_2$.
* $mn=2$: angle $3\pi/4$ and squared-length ratio $2$, giving $B_2$.
* $mn=3$: angle $5\pi/6$ and squared-length ratio $3$, giving $G_2$.
The root strings generated by the two simple reflections produce exactly the roots in those four standard systems. Hence these are all possibilities, proving the <classification of rank-two root systems>.
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