Toward the classification of normalized integral table algebras generated by a faithful non-real element of degree 3 with (I)

The classification of integral table algebras (A, B) generated by a faithful element of degree 2 with (Formula presented.) was completed by Blau in 1995. After that, it is an interesting topic to classify integral table algebras generated by a faithful non-real element of degree 3 with (Formula presented.) But it is difficult, so Arad and Chen et al. in 2005 and 2011 studied normalized integral table algebras generated by a faithful non-real element of degree 3 by assuming (Formula presented.) and (Formula presented.) they almost classified such table algebras. Here the authors continue this work but not assuming (Formula presented.) Communicated by Miriam Cohen.