We compute the total Chow-Witt rings of the classifying space Bµ_n of the roots of unity, as well as the products BG_m x Bµ_n and Bµ_m x Bµ_n for all m and n greater than or equal to 1 based on the strategy by di Lorenzo and Mantovani (2023) for Bµ_n with n even. Moreover we compute the total I^j-cohomology and Chow-Witt rings of the product of two projective spaces P^q x P^r for all q and r greater than or equal to 1, and of the product BG_m x BG_m.
