= Solution
The <Fundamental representations of B2> have weight sets
$$
d_{[1,0]}:\quad
\left\{\left(\frac12,\frac12\right),
\left(\frac12,-\frac12\right),
\left(-\frac12,\frac12\right),
\left(-\frac12,-\frac12\right)\right\},
$$
so $\dim d_{[1,0]}=4$, and
$$
d_{[0,1]}:\quad
\{(1,0),(-1,0),(0,1),(0,-1),(0,0)\},
$$
so $\dim d_{[0,1]}=5$. The <Adjoint representation> has all eight roots as nonzero weights and zero with multiplicity two. Its <highest root> is
$$
2\alpha_1+\alpha_2=(1,1)=2\omega_1,
$$
so its highest-weight label is $[2,0]$ and its dimension is $8+2=10$.
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