For $B_3\cong\mathfrak{so}_7(\mathbb C)$, the highest weight $[0,0,1]$ is the <spinor representation>. Its eight weights, all of multiplicity one, are
$$
\begin{gathered}
[0,0,1],\ [0,1,-1],\ [1,-1,1],\ [1,0,-1],\\
[-1,0,1],\ [-1,1,-1],\ [0,-1,1],\ [0,0,-1].
\end{gathered}
$$
Equivalently, in an orthonormal basis they are $(\pm e_1\pm e_2\pm e_3)/2$ with independent signs. Hence $\dim d_3=8$.

Solution

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

{wiki=Spin_representation}

A spinor representation is a representation of a <Spin group>, or infinitesimally of a <Special orthogonal Lie algebra>, constructed from a <Clifford algebra>. In even dimensions it can split into two chiral spinor representations.

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