Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 3 / b / iv / Solution

The linear mappreserves $\Phi$ and exchanges ${\alpha }_2$ with ${\alpha }_3$ while fixing ${\alpha }_1$. It is not in the $D_3$ Weyl group, whose signed permutations change an even number of signs. Thus it is an outer automorphism of the root system.