Fermion masses for the Englert solution on the round seven-sphere

Using a mass formula for Englert-type solutions on arbitrary coset spaces, we have found that if the coset is the round $S_7$, no massless fermions exist. For this purpose we develop a method to decompose SO(8) representations with respect to spin(7), which enables us to compute the eigenvalues of the spin(7)-invariant differential operators in the mass matrix by using algebraic identities satisfied by them.