Homogeneous spin Riemannian manifolds with the simplest Dirac operator

We show the existence of nonsymmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds $(M,g)$ which are traceless cyclic with respect to some quotient expression $M=G\mathrm{/}K$ and reductive decomposition $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{m}$. Using transversally symmetric fibrations of noncompact type, we give a list of them.