Abstract
It is well known that for a quaternion algegra, the anisotropy of its norm form determines if the quaternion algebra is a division algebra. In case of biquaternio algebra, the anisotropy of the associated Albert form (as defined in [LLT]) determines if the biquaternion algebra is a division ring. In 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 natural way to associate a tensor product of quaternion algebras with a form such that when the involution is orthogonal, the algebra is a Baer ordered *-field iff the associated form is anisotropic.
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Leung, K.H. Forms and Baer ordered *-fields. Isr. J. Math. 116, 1–19 (2000). https://doi.org/10.1007/BF02773209
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DOI: https://doi.org/10.1007/BF02773209