Editing Split-quaternion (section)

=== Stratification by the norm ===
As seen above, the purely nonreal split-quaternions of norm {{math|–1, 1}} and {{math|0}} form respectively a hyperboloid of one sheet, a hyperboloid of two sheets and a [[circular cone]] in the space of non real quaternions.

These surfaces are pairwise [[asymptote]] and do not intersect. Their [[set complement|complement]] consist of six connected regions:
* the two regions located on the concave side of the hyperboloid of two sheets, where <math>N(q)>1</math>
* the two regions between the hyperboloid of two sheets and the cone, where <math>0<N(q)<1</math>
* the region between the cone and the hyperboloid of one sheet where <math>-1<N(q)<0</math>
* the region outside the hyperboloid of one sheet, where <math>N(q)<-1</math>

This stratification can be refined by considering split-quaternions of a fixed norm: for every real number {{math|''n'' ≠ 0}} the purely nonreal split-quaternions of norm {{math|''n''}} form an hyperboloid. All these hyperboloids are asymptote to the above cone, and none of these surfaces intersect any other. As the set of the purely nonreal split-quaternions is the [[disjoint union]] of these surfaces, this provides the desired stratification.