 Solution

Map the quaternion generators to $i{\sigma }_x,i{\sigma }_y$ (and their products), obtaining the unique two-dimensional irreducible $V$ and an embedding $Q_8\subset SU(2)$. Clebsch-Gordan gives $V\otimes V\simeq {\mathrm{Sym}}^{2}V\oplus 1$ and $V^{\otimes 3}\simeq {\mathrm{Sym}}^{3}V\oplus V\oplus V$. On restriction, ${\mathrm{Sym}}^{2}V$ is the sum of the three nontrivial one-dimensional characters of $Q_8$, while the other summand is trivial.