The chain lemma for biquaternion algebras

Abstract

Any two decompositions of a biquaternion algebra over a field F into a sum of two quaternion algebras can be connected by a chain of decompositions such that any two neighboring decompositions are ( a , b ) + ( c , d ) and ( a c , b ) + ( c , b d ) for some a , b , c , d ∈ F ⁎ . A similar result is established for decompositions of a biquaternion algebra into a sum of three quaternions if F has no cubic extension.