Non-existence of imprimitive QQ-polynomial schemes of exceptional type with d=4d=4

In [H. Suzuki, Imprimitive QQ-polynomial association schemes, J. Algebraic Combin. 7 (2) (1998) 165–180], it was shown that an imprimitive QQ-polynomial scheme X=(X,{Ri}0≤i≤d)X=(X,{Ri}0≤i≤d) is either dual bipartite, dual antipodal or of class 4 or 6. In this paper, it will be shown that the scheme of class 4 does not occur using the integrality conditions of the entries of the first eigenmatrix of XX. These integrality conditions arise from the fact that XX has exactly one QQ-polynomial ordering [H. Suzuki, Association schemes with multiple QQ-polynomial structures, J. Algebraic Combin. 7 (2) (1998) 181–196].