C3 root system ($C_3$)

The $C_3$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $1\le i<j\le 3$ and $\pm 2{\epsilon }_i$ for $1\le i\le 3$. One root basis is ${\alpha }_1={\epsilon }_1-{\epsilon }_2$, ${\alpha }_2={\epsilon }_2-{\epsilon }_3$, ${\alpha }_3=2{\epsilon }_3$. With $A_{ij}=⟨{\alpha }_i,{\alpha }_j^{\vee }⟩$, its Cartan matrix isThe fundamental representation of highest weight $[1,0,0]$ is six-dimensional, with weights $\pm {\epsilon }_1,\pm {\epsilon }_2,\pm {\epsilon }_3$.