The Albert quadratic form for an algebra of degree four

Abstract:

Suppose $K$ is a field and the $K$-algebra $A$ is expressed as a tensor product of two quaternion algebras $A \cong {H_1} \otimes {H_2}$. Let ${N_i}$ be the norm form on ${H_i}$ and define the "Albert form" ${\alpha _A}$ to be the $6$-dimensional quadratic form determined by $\alpha _{A} \bot \left \langle {1, - 1} \right \rangle \cong {N_1} \bot - {N_2}$. In [Adv. in Math. 48 (1983), 149-165] Jacobson proved: (1) any two Albert forms for $A$ are similar; (2) if $A$ and $B$ are algebras of this type, then $A \cong B$ if and only if ${\alpha _A}$ and ${\alpha _B}$ are similar. The authors prove this result using quadratic forms and Clifford algebras, avoiding the application of Jacobson’s theory of Jordan norms.