Forms and Baer ordered *-fields

It is well known that for a quaternion algebra, the anisotropy of its norm form determines if the quaternion algebra is a division algebra. In case of biquaternion algebra, the anisotropy of the associated Albert form (as def ined in [LLT]) determines if the biquaternion algebra is a division ring. I n these situations, the norm forms and the Albert forms are quadratic forms over the center of the quaternion algebras; and they are strongly related to the algebraic structure of the algebras. As it turns out, there is a nat ural way to associate a tensor product of quaternion algebras with a form s uch that when the involution is orthogonal, the algebra is a Baer ordered * -field iff the associated form is anisotropic.