= B3 root system
{c}
{title2=$B_3$}
The $B_3$ roots are $\pm\varepsilon_i\pm\varepsilon_j$ for $1\leq i<j\leq3$ and $\pm\varepsilon_i$ for $1\leq i\leq3$. Its root basis $\alpha_1=\varepsilon_1-\varepsilon_2$, $\alpha_2=\varepsilon_2-\varepsilon_3$, $\alpha_3=\varepsilon_3$ exhibits it as the <dual root system> of the <C3 root system>. With $A_{ij}=\langle\alpha_i,\alpha_j^\vee\rangle$, its <Cartan matrix> is
$$
\begin{pmatrix}2&-1&0\\-1&2&-2\\0&-1&2\end{pmatrix}.
$$
The fundamental representation of highest weight $[1,0,0]$ is the seven-dimensional vector representation, with weights $\pm\varepsilon_1,\pm\varepsilon_2,\pm\varepsilon_3,0$.
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