On the Dirac spectrum of homogeneous 3-spheres

We show that any two left-invariant metrics on $S^3\cong \mathrm{SU}(2)$ which are isospectral for the associated classical Dirac operator $D$ must be isometric. In the case of left-invariant metrics of positive scalar curvature, we compute and use the smallest eigenvalue of $D^2$. We show analogous results for left-invariant metrics on $\mathrm{SO}(3)=S^3\mathrm{/}\{\pm 1\}$ for each of its two spin structures.