  We study the decomposition as an $\textrm{SO}(3)$-module of the multiplicity
space corresponding to the branching from $\textrm{SO}(n+3)$ to
$\textrm{SO}(n)$. Here, $\textrm{SO}(n)$ (resp.\ $\textrm{SO}(3)$) is
considered embedded in $\textrm{SO}(n+3)$ in the upper left-hand block (resp.\
lower right-hand block). We show that when the highest weight of the
irreducible representation of $\textrm{SO}(n)$ interlaces the highest weight of
the irreducible representation of $\textrm{SO}(n+3)$, then the multiplicity
space decomposes as a tensor product of $\lfloor (n+2)/2\rfloor$ reducible
representations of $\textrm{SO}(3)$.
