Editing Split-quaternion (section)

=== Hyperbolic units ===
[[Image:HyperboloidOfOneSheet.PNG|right|thumb|Hyperboloid of one sheet, source of [[hyperbolic unit]]s.<br>(the vertical axis is called {{mvar|x}} in the article)]]
This is the case where {{math|''N''(''q'') < 0}}. Letting <math display="inline">n=\sqrt{-N(q)},</math>  one has
: <math>q^2 = -q^*q=N(q)=-n^2=x^2-y^2-z^2.</math>
It follows that {{math|{{sfrac|''n''}} ''q''}} belongs to the [[hyperboloid of one sheet]] of equation {{math|1=''y''<sup>2</sup> + ''z''<sup>2</sup> − ''x''<sup>2</sup> = 1}}. Therefore, there are real numbers {{math|''n'', ''t'', ''u''}} such that {{math|0 ≤ ''t'' < 2{{pi}}}} and 
: <math>p=w+n\sinh(u)\mathrm i + n\cosh(u)\cos(t)\mathrm j + n\cosh(u)\sin(t)\mathrm k.</math>

This is a parametrization of all split-quaternions whose nonreal part has a negative norm.

This is also a parameterization of the corresponding subalgebras by the pairs of opposite points of a hyperboloid of one sheet: the split-quaternions of the form <math>\sinh(u)\mathrm i + \cosh(u)\cos(t)\mathrm j + \cosh(u)\sin(t)\mathrm k</math> form a hyperboloid of one sheet; a subalgebra generated by a split-quaternion with a nonreal part of negative norm contains exactly two opposite points on this hyperboloid; and the hyperboloid does not contain any other point.

The algebra generated by a split-quaternion with a nonreal part of negative norm is isomorphic to <math>\mathbb R[X]/\langle X^2-1\rangle</math> and to the [[ring (mathematics)|ring]] of [[split-complex number]]s. It is also isomorphic (as an algebra) to <math>\mathbb R^2</math> by the mapping defined by
<math display="inline">(1,0)\mapsto \frac{1+X}2, \quad
(0,1)\mapsto \frac{1-X}2.
</math>