Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 3 / e / Solution

For equal-length candidate roots, the required inner products are

Set A is invalid because ${\alpha }_1\cdot {\alpha }_3=-1$ rather than zero; indeed its three vectors sum to zero and are not linearly independent. Set B has all the displayed inner products and is linearly independent, so it is valid. Set C is the standard realizationof the A3 root system and is also valid.