Since each coefficient w, x, y, z spans two real dimensions, the split-biquaternion is an element of an eight-dimensional vector space.
This algebra was introduced by William Kingdon Clifford in an 1873 article for the London Mathematical Society.
It has been repeatedly noted in mathematical literature since then, variously as a deviation in terminology, an illustration of the tensor product of algebras, and as an illustration of the direct sum of algebras.
This is the geometric algebra generated by three orthogonal imaginary unit basis directions, {e1, e2, e3} under the combination rule giving an algebra spanned by the 8 basis elements {1, e1, e2, e3, e1e2, e2e3, e3e1, e1e2e3}, with (e1e2)2 = (e2e3)2 = (e3e1)2 = −1 and ω2 = (e1e2e3)2 = +1.
The sub-algebra spanned by the 4 elements {1, i = e1, j = e2, k = e1e2} is the division ring of Hamilton's quaternions, H = Cl0,2(R).
Equivalently, The split-biquaternions form an associative ring as is clear from considering multiplications in its basis {1, ω, i, j, k, ωi, ωj, ωk}.
But split-complex numbers form a ring, not a field, so vector space is not appropriate.
This standard term of ring theory expresses a similarity to a vector space, and this structure by Clifford in 1873 is an instance.
Though split-biquaternions form an eight-dimensional space like Hamilton's biquaternions, on the basis of the Proposition it is apparent that this algebra splits into the direct sum of two copies of the real quaternions.
The split-biquaternions should not be confused with the (ordinary) biquaternions previously introduced by William Rowan Hamilton.