B2 root system ($B_2$)

The $B_2$ root system consists of $\pm {\epsilon }_1$, $\pm {\epsilon }_2$, and $\pm {\epsilon }_1\pm {\epsilon }_2$. Its vector representation has weights ${\epsilon }_1,{\epsilon }_2,0,-{\epsilon }_2,-{\epsilon }_1$.

The root systems $B_2$ and $C_2$ are isomorphic after interchanging long and short simple-root labels. The classification of complex simple Lie algebras therefore gives ${\mathfrak{so}}_5(\mathbb{C})\cong {\mathfrak{sp}}_4(\mathbb{C})$.

For simple roots $(1,0)$ and $(-1,1)$, the fundamental weights are ${\omega }_1=(1/2,1/2)$ and ${\omega }_2=(0,1)$. The representation $[1,0]$ is the four-dimensional spin representation with weights $(\pm 1/2,\pm 1/2)$; $[0,1]$ is the five-dimensional vector representation with weights $(\pm 1,0),(0,\pm 1),(0,0)$.