Төрттіктер мен бөлінген кешен сандарды қолданатын алгебра элементі

Element of an algebra using quaternions and split complex numbers
In mathematics, a split biquaternion is a hypercomplex number of the form

where w, x, y, and z are split complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient w, x, y, z spans two real dimensions, the split biquaternion is an element of an eight dimensional vector space. Considering that it carries a multiplication, this vector space is an algebra over the real field, or an algebra over a ring where the split complex numbers form the ring. 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. The split biquaternions have been identified in various ways by algebraists; see below.

The split biquaternions form an associative ring as is clear from considering multiplications in its basis When ω is adjoined to the quaternion group one obtains a 16 element group
( {1, i, j, k, −1, −i, −j, −k, ω, ωi, ωj, ωk, −ω, −ωi, −ωj, −ωk}, × ).

Since elements of the quaternion group can be taken as a basis of the space of split biquaternions, it may be compared to a vector space. But split complex numbers form a ring, not a field, so vector space is not appropriate. Rather the space of split biquaternions forms a free module. This standard term of ring theory expresses a similarity to a vector space, and this structure by Clifford in 1873 is an instance. Split biquaternions form an algebra over a ring, but not a group ring.

The split biquaternions should not be confused with the (ordinary) biquaternions previously introduced by William Rowan Hamilton. Hamilton's biquaternions are elements of the algebra