Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 102 / 2 / a / Solution

Write an element of the diagonal Cartan subalgebra asand define the linear functionals ${\epsilon }_i(t)=t_i$. The root-space decomposition of the Symplectic Lie algebra ${\mathfrak{sp}}_6(\mathbb{C})$ then has the C3 root system