Simple $B_4$ representations associated to cyclotomic Hecke algebras

We determine the structure of the cyclotomic Hecke algebra corresponding to the complex reflection group $G_{25}$ also when it is not semisimple, as long as the generators are diagonalizable. In particular, we classify all simple representations of the braid group $B_4$ for which the generators are diagonalizable and satisfy a cubic polynomial. This will be used in the classification of braided tensor categories of type $G_2$.