The biquaternions of William Rowan Hamilton (1844) and the related split-biquaternions and dual quaternions do not form biquaternion algebras in this sense.
[1][2] A biquaternion algebra is a central simple algebra of dimension 16 and degree 4 over the base field: it has exponent (order of its Brauer class in the Brauer group of F)[3] equal to 1 or 2.
Let A = (a1,a2) and B = (b1,b2) be quaternion algebras over F. The Albert form for A, B is It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of A and B.
[4] The quaternion algebras are linked if and only if the Albert form is isotropic, otherwise unlinked.
[5] Albert's theorem states that the following are equivalent: In the case of linked algebras we can further classify the other possible structures for the tensor product in terms of the Albert form.