On the SO(n+3) to SO(n) branching multiplicity space

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