Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 4 / ii / 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 $mn$:

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.