Certain properties of the tensor product of two Albert forms

Abstract

Let F be a field, Let further D1, D2 be biquaternion algebras over F with Albert forms We prove that the dimension of anisotropic part of the form is at most 24, provided there exists a common splitting field of degree for D1 and D2. We also investigate the symbol length of i.e., the minimal number n such that is the sum of n 4-fold Pfister forms over F modulo It turns out that if is a biquaternion algebra as well, then and this inequality is strict. Among negative results we give examples of biquaternion division algebras D1, D2 without common quadratic subfield, and such that is a biquaternion algebra.