Editing Split-quaternion (section)

=== Imaginary units ===
[[Image:HyperboloidOfTwoSheets.svg|right|thumb|Hyperboloid of two sheets, source of [[imaginary unit]]s]]
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 two sheets]] of equation <math>x^2-y^2-z^2=1.</math> Therefore, there are real numbers {{math|''n'', ''t'', ''u''}} such that {{math|0 ≤ ''t'' < 2{{pi}}}} and 
: <math>p=w+n\cosh(u)\mathrm i + n\sinh(u)\cos(t)\mathrm j + n\sinh(u)\sin(t)\mathrm k.</math>

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

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

The algebra generated by a split-quaternion with a nonreal part of positive norm is isomorphic to <math>\mathbb R[X]/\langle X^2+1\rangle</math> and to the field <math>\Complex</math> of [[complex number]]s.