Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 4 / b

With $A_{ij}=2({\alpha }_i,{\alpha }_j)/({\alpha }_j,{\alpha }_j)$ and roots labeled from left to right, the three Cartan matrices are

All roots of $A_3$ have the same length, so $|{\alpha }_2|/|{\alpha }_3|=1$. In $B_3$, ${\alpha }_2$ is long and ${\alpha }_3$ short, giving $\sqrt{2}$. In $C_3$ their roles are reversed, giving $1/\sqrt{2}$. These ratios also follow from $A_{23}/A_{32}=|{\alpha }_2{|}^2/|{\alpha }_3{|}^2$.

Disconnected nodes in a Dynkin diagram represent orthogonal roots, so $\angle ({\alpha }_1,{\alpha }_3)=90^{\circ }$ in all three cases. A single edge gives $120^{\circ }$, hence $\angle ({\alpha }_1,{\alpha }_2)=120^{\circ }$ and, for $A_3$, also $\angle ({\alpha }_2,{\alpha }_3)=120^{\circ }$. The double edge in both $B_3$ and $C_3$ gives $\angle ({\alpha }_2,{\alpha }_3)=135^{\circ }$.