Geodesic
Clifford algebras, Möbius transformations, Vahlen matrices, and $B$-loops
Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 2, pp. 319-331.

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In this paper we show that well-known relationships connecting the Clifford algebra on negative euclidean space, Vahlen matrices, and Möbius transformations extend to connections with the Möbius loop or gyrogroup on the open unit ball $B$ in $n$-dimensional euclidean space $\mathbb R^n$. One notable achievement is a compact, convenient formula for the Möbius loop operation $a\ast b=(a+b)(1-ab)^{-1}$, where the operations on the right are those arising from the Clifford algebra (a formula comparable to $(w+z)(1+\overline wz)^{-1}$ for the Möbius loop multiplication in the unit complex disk).
Classification : 15A66, 20N05, 51B10
Keywords: Bruck loop; Clifford algebra; gyrogroup; Möbius transformations; Vahlen matrices; involutive group 
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     author = {Lawson, Jimmie},
     title = {Clifford algebras, {M\"obius} transformations, {Vahlen} matrices, and $B$-loops},
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     zbl = {1211.20064},
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     url = {http://geodesic.mathdoc.fr/item/CMUC_2010__51_2_a15/}
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Lawson, Jimmie. Clifford algebras, Möbius transformations, Vahlen matrices, and $B$-loops. Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 2, pp. 319-331. http://geodesic.mathdoc.fr/item/CMUC_2010__51_2_a15/