Editing Split-quaternion (section)

== Properties ==

Like the [[quaternion]]s introduced by [[William Rowan Hamilton|Hamilton]] in 1843, they form a four [[dimension (vector space)|dimensional]] real [[associative algebra]]. But like the real algebra of 2×2 matrices – and unlike the real algebra of quaternions – the split-quaternions contain nontrivial [[zero divisor]]s, [[nilpotent]] elements, and [[idempotent element (ring theory)|idempotent]]s. (For example, {{nowrap|{{sfrac|1|2}}(1 + j)}} is an idempotent zero-divisor, and {{nowrap|i − j}} is nilpotent.)  As an [[algebra over a field|algebra over the real numbers]], the algebra of split-quaternions is [[algebra isomorphism|isomorphic]] to the algebra of 2×2 real matrices by the above defined isomorphism. 

This isomorphism allows identifying each split-quaternion with a 2×2 matrix. So every property of split-quaternions corresponds to a similar property of matrices, which is often named differently. 

The ''conjugate'' of a split-quaternion
{{math|1=''q'' = ''w'' + ''x''i + ''y''j + ''z''k}}, is {{math|1=''q''<sup>∗</sup> = ''w'' − ''x''i − ''y''j − ''z''k}}. In term of matrices, the conjugate is the [[cofactor matrix]] obtained by exchanging the diagonal entries and changing the sign of the other two entries.

The product of a split-quaternion with its conjugate is the [[isotropic quadratic form]]:
: <math>N(q) = q q^* = w^2 + x^2 - y^2 - z^2,</math>
which is called the [[Norm (mathematics)#Composition algebras|''norm'']] of the split-quaternion or the [[determinant]] of the associated matrix.

The real part of a split-quaternion {{math|1=''q'' = ''w'' + ''x''i + ''y''j + ''z''k}} is {{math|1=''w'' = (''q''<sup>∗</sup> + ''q'')/2}}. It equals the [[trace (linear algebra)|trace]] of associated matrix.

The norm of a product of two split-quaternions is the product of their norms. Equivalently, the determinant of a product of matrices is the product of their determinants. This property means that split-quaternions form a [[composition algebra]]. As there are nonzero split-quaternions having a zero norm, split-quaternions form a "split composition algebra" – hence their name.

A split-quaternion with a nonzero norm has a [[multiplicative inverse]], namely {{math|''q''<sup>∗</sup>/''N''(''q'')}}. In terms of matrices, this is equivalent to the [[Cramer rule]] that asserts that a matrix is [[invertible matrix|invertible]] if and only its determinant is nonzero, and, in this case, the inverse of the matrix is the quotient of the cofactor matrix by the determinant.

The isomorphism between split-quaternions and 2×2 real matrices shows that the multiplicative group of split-quaternions with a nonzero norm is isomorphic with <math>\operatorname{GL}(2, \mathbb R),</math> and the group of split quaternions of norm {{math|1}} is isomorphic with <math>\operatorname{SL}(2, \mathbb R).</math>

Geometrically, the split-quaternions can be compared to Hamilton's quaternions as [[pencil of planes|pencils of planes]]. In both cases the real numbers form the axis of a pencil. In Hamilton quaternions there is a sphere of imaginary units, and any pair of antipodal imaginary units generates a complex plane with the real line. For split-quaternions there are hyperboloids of hyperbolic and imaginary units that generate split-complex or ordinary complex planes, as described below in [[#Stratification|§ Stratification]].