Editing Split-quaternion (section)

=== Nilpotent case ===
With above notation, if <math>q^2=0,</math> (that is, if {{math|''q''}} is [[nilpotent]]), then {{math|1=''N''(''q'') = 0}}, that is, <math>x^2-y^2-z^2=0.</math>  This implies that there exist {{mvar|w}} and {{mvar|t}} in <math>\mathbb R</math> such that {{math|0 ≤ ''t'' < 2{{pi}}}} and
: <math>p=w+a\mathrm i + a\cos(t)\mathrm j + a\sin(t)\mathrm k.</math>

This is a parametrization of all split-quaternions whose nonreal part is nilpotent. 

This is also a parameterization of these subalgebras by the points of a circle: the split-quaternions of the form <math>\mathrm i + \cos(t)\mathrm j + \sin(t)\mathrm k</math> form a [[circle]]; a subalgebra generated by a nilpotent element contains exactly one point of the circle; and the circle does not contain any other point.  

The algebra generated by a nilpotent element is isomorphic to <math>\mathbb R[X]/\langle X^2\rangle</math> and to the plane of [[dual number]]s.