Geodesic
The space $H_4$ and quaternion algebra
Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 23 (1997), pp. 187-198
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We obtain a conformal model of the four-dimensional Lobachevskii space $H_4$ by an autopolar framing of a quadric in the projective space $P_5$. We use quaternions to describe points and vectors, this allows us to write the parallel translation law in terms of quaternions. In these terms we also represent infinitesimal motions and infinitesimal conformal transformations, and some finite transformations of $H_4$ as well. 
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     author = {A. P. Shirokov},
     title = {The space $H_4$ and quaternion algebra},
     journal = {Trudy Geometricheskogo Seminara},
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A. P. Shirokov. The space $H_4$ and quaternion algebra. Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 23 (1997), pp. 187-198. http://geodesic.mathdoc.fr/item/KUTGS_1997_23_a16/