Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 302 / 1 / i / Solution

The squared root lengths and inner product areWith $A_{ij}=2{\alpha }_i\cdot {\alpha }_j/|{\alpha }_j{|}^2$, the Cartan matrix isThis is the Cartan matrix of the G2 root system, with ${\alpha }_1$ short and ${\alpha }_2$ long.