Solution
= Solution
Write an element of the diagonal <Cartan subalgebra> as
$$
t=\operatorname{diag}(t_1,t_2,t_3,-t_1,-t_2,-t_3)
$$
and define the <linear functionals> $\varepsilon_i(t)=t_i$. The <root-space decomposition> of the <Symplectic Lie algebra> $\mathfrak{sp}_6(\mathbb C)$ then has the <C3 root system>
$$
\boxed{\Phi=\{\pm\varepsilon_i\pm\varepsilon_j:1\leq i<j\leq3\}\cup\{\pm2\varepsilon_i:1\leq i\leq3\}.}
$$