algebra generated by a split-quaternion with a nonreal part of positive norm is isomorphic to and to the plane of split-complex numbers. It is also isomorphic (as an algebra) to by the mapping defined by Indecomposable case This is the case where . Letting one has It follows that belongs to the hyperboloid of one sheet of equation Therefore, there are real numbers such that and 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 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 and to field of complex numbers. Stratification by the norm As seen above, the purely nonreal split-quaternions of norm and 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 complement consist of six connected regions: the two regions located on the concave side of the hyperboloid of two sheets, where the two regions between the hyperboloid of two sheets and the cone, where the region between the cone and the hyperboloid of one sheet where the region outside the hyperboloid of one sheet, where This stratification can be refined by considering split-quaternions of a fixed norm: for every real number the purely nonreal split-quaternions of norm 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. Historical notes The coquaternions were initially introduced (under that name) in 1849 by James Cockle in the London–Edinburgh–Dublin Philosophical Magazine. The introductory papers by Cockle were recalled in the 1904 Bibliography of the Quaternion Society. Alexander Macfarlane called the structure of split-quaternion vectors an exspherical system when he was speaking at the International Congress of Mathematicians in Paris in 1900. The unit sphere was considered in 1910 by Hans Beck. For example,