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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 the ''Synonyms'' section below. ==Modern definition== A split-biquaternion is a member of the Clifford algebra ''C''ℓ0,3(R). This is the geometric algebra generated by three orthogonal imaginary unit basis directions, under the combination rule :: giving an algebra spanned by the 8 basis elements , with (''e''1''e''2)2 = (''e''2''e''3)2 = (''e''3''e''1)2 = −1 and (ω = ''e''1''e''2''e''3)2 = +1. The sub-algebra spanned by the 4 elements is the division ring of Hamilton's quaternions, H = ''C''ℓ0,2(R). One can therefore see that : where D = ''C''ℓ1,0(R) is the algebra spanned by , the algebra of the split-complex numbers. Equivalently, : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Split-biquaternion」の詳細全文を読む スポンサード リンク
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