geometry for describing hyperbolic motions of the Poincaré disk model. Generation from split-complex numbers Split-quaternions may be generated by modified Cayley-Dickson construction similar to the method of L. E. Dickson and Adrian Albert. for the division algebras C, H, and O. The multiplication rule is used when producing the doubled product in the real-split cases. The doubled conjugate so that If a and b are split-complex numbers and split-quaternion then Stratification In this section, the subalgebras generated by a single split-quaternion are studied and classified. Let be a split-quaternion. Its real part is . Let be its nonreal part. One has , and therefore It follows that is a real number if and only is either a real number ( and ) or a purely nonreal split quaternion ( and ). The structure of the subalgebra generated by follows straightforwardly. One has and this is a commutative algebra. Its dimension is two except if is real (in this case, the subalgebra is simply ). The nonreal elements of whose square is real have the form with Three cases have to be considered, which are detailed in the next subsections. Nilpotent case With above notation, if (that is, if is nilpotent), then , that is, This implies that there exist and in such that and 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 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 and to the plane of dual numbers. Decomposable case This is the case where . Letting one has It follows that belongs to the hyperboloid of two sheets of equation Therefore, there are real numbers such that and 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 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