On some Hermitian variations of Hodge structure of Calabi-Yau type with real multiplication

We prove that, for every totally real number field E_0, there exists a weight three variation of Hodge structure of Calabi-Yau type defined over the rational numbers with associated endomorphism algebra E_0 such that the unique irreducible factor of Calabi-Yau type of the corresponding real variation of Hodge structure is the canonical real VHS of CY type over the Hermitian symmetric domain II_6, associated to the real group SO^*(12). The main point is a rationality result for the half spin representations of a form of the group SO^*(4m) defined over a number field.