= Solution
Using $\mathfrak{so}_4\cong\mathfrak{su}_2\oplus\mathfrak{su}_2$, the five-dimensional vector representation restricts as
$$
d_{[0,1]}\to(\tfrac12,\tfrac12)\oplus(0,0),
$$
which is the $\mathbf4\oplus\mathbf1$ decomposition from part (c). The four-dimensional spin representation is naturally a representation of $\mathfrak{so}_5$ or $Spin(5)$ rather than an honest representation of $SO(5)$; it restricts as
$$
d_{[1,0]}\to(\tfrac12,0)\oplus(0,\tfrac12),
$$
the two chiral spin representations of $\mathfrak{so}_4$. Finally,
$$
d_{[2,0]}\to(1,0)\oplus(0,1)\oplus(\tfrac12,\tfrac12),
$$
namely the two three-dimensional summands of the $SO(4)$ adjoint plus its four-dimensional vector, agreeing with $\mathbf{10}\to\mathbf6\oplus\mathbf4$ from the <SO5 to SO4 branching> calculation.
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