Let  be an algebraic number field of degree d and discriminant  over  Let  be an associative algebra over  given by structure constants such that  holds for some positive integer n. Suppose that d, n and are bounded. In a previous paper, a polynomial time ff-algorithm was given to construct explicitly an isomorphism

Here we simplify and improve this algorithm in the cases    and  with  or  The improvements are based on work by Y. Kitaoka and R. Coulangeon on tensor products of lattices.

central simple algebra, maximal order, real and complex embedding, lattice basis reduction, tensor product of lattices, Hermite constant, Bergé-Martinet constant.