The chain lemma for biquaternion algebras

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 (ac,b)+(c,bd) 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.