Bn root system ($B_n$)

In an orthonormal basis $({\epsilon }_i)$, the $B_n$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $i<j$ and $\pm {\epsilon }_i$.

For $n\ge 2$, the Bn root system has simple roots ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$ for $i<n$ and ${\alpha }_n={\epsilon }_n$. Its Dynkin diagram is a chain with a double terminal bond pointing toward short ${\alpha }_n$. The highest root is $\theta ={\epsilon }_1+{\epsilon }_2$. In the Extended Dynkin diagram, ${\alpha }_0=-\theta$ attaches by a single bond to ${\alpha }_2$ for $n\ge 3$. For $n=2$, both long nodes $0,1$ attach by double bonds to short node $2$. Rank one is the rank-one root system; its affine diagram has two equal-length nodes joined by a double bond without a short-root arrow.

The $B_3$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $1\le i<j\le 3$ and $\pm {\epsilon }_i$ for $1\le i\le 3$. Its root basis ${\alpha }_1={\epsilon }_1-{\epsilon }_2$, ${\alpha }_2={\epsilon }_2-{\epsilon }_3$, ${\alpha }_3={\epsilon }_3$ exhibits it as the dual root system of the C3 root system. With $A_{ij}=⟨{\alpha }_i,{\alpha }_j^{\vee }⟩$, its Cartan matrix isThe fundamental representation of highest weight $[1,0,0]$ is the seven-dimensional vector representation, with weights $\pm {\epsilon }_1,\pm {\epsilon }_2,\pm {\epsilon }_3,0$.