Solution
= Solution
One root basis is
$$
\alpha_1=\varepsilon_1-\varepsilon_2,
\qquad
\alpha_2=\varepsilon_2-\varepsilon_3,
\qquad
\alpha_3=\varepsilon_2+\varepsilon_3.
$$
All roots have the same length. The nonzero inner products are $(\alpha_1,\alpha_2)=(\alpha_1,\alpha_3)=-1$, so the labeled <Dynkin diagram> is
$$
\alpha_2\;\text{---}\;\alpha_1\;\text{---}\;\alpha_3.
$$
It is the three-node $A_3$ diagram.