the dihedral group appears on page 419. The split-quaternion structure has also been mentioned briefly in the Annals of Mathematics. In 1995 Ian Porteous placed split-quaternions in the context of Clifford algebra and hypercomplex numbers. Synonyms Para-quaternions (Ivanov and Zamkovoy 2005, Mohaupt 2006) Manifolds with para-quaternionic structures are studied in differential geometry and string theory. In the para-quaternionic literature k is replaced with −k. Exspherical system (Macfarlane 1900) Split-quaternions (Rosenfeld 1988) Antiquaternions (Rosenfeld 1988) Pseudoquaternions (Yaglom 1968 Rosenfeld 1988) See also Pauli matrices Split-biquaternions Split-octonions Dual quaternions Notes Further reading Brody, Dorje C., and Eva-Maria Graefe. "On complexified mechanics and coquaternions." Journal of Physics A: Mathematical and Theoretical 44.7 (2011): 072001. Ivanov, Stefan; Zamkovoy, Simeon (2005), "Parahermitian and paraquaternionic manifolds", Differential Geometry and its Applications 23, pp. 205–234, , . Mohaupt, Thomas (2006), "New developments in special geometry", . Özdemir, M. (2009) "The roots of a split quaternion", Applied Mathematics Letters 22:258–63. Özdemir, M. & A.A. Ergin (2006) "Rotations with timelike quaternions in Minkowski 3-space", Journal of Geometry and Physics 56: 322–36. Pogoruy, Anatoliy & Ramon M Rodrigues-Dagnino (2008) Some algebraic and analytical properties of coquaternion algebra, Advances in Applied Clifford Algebras. Composition algebras Quaternions Hyperbolic geometry Special relativity