Abstract: We compute the fundamental Dirac operator for the three-parameter-family of homogeneous Riemannian metrics and the four different spin structures on $\raise{0.5ex}{\mathrm{SU}_2}\!/\!\raise{-0.5ex}{\mathrm{Q}_8}$, where $\mathrm{Q}_8$ denotes the group of quaternions. We deduce its spectrum for the Berger metrics and show the sharpness of Christian Bär's upper bound for the smallest Dirac eigenvalue in the particular case where $\raise{0.5ex}{\mathrm{SU}_2}\!/\!\raise{-0.5ex}{\mathrm{Q}_8}$ is a homogeneous minimal hypersurface of $S^4$.