Ca. Manogue et J. Schray, FINITE LORENTZ TRANSFORMATIONS, AUTOMORPHISMS, AND DIVISION-ALGEBRAS, Journal of mathematical physics, 34(8), 1993, pp. 3746-3767
An explicit algebraic description of finite Lorentz transformations of
vectors in ten-dimensional Minkowski space is given by means of a par
ametrization in terms of the octonions. The possible utility of these
results for superstring theory is mentioned. Along the way automorphis
ms of the two highest dimensional normed division algebras, namely, th
e quaternions and the octonions, are described in terms of conjugation
maps. Similar techniques are used to define SO(3) and SO(7) via conju
gation, SO(4) via symmetric multiplication, and SO(8) via both symmetr
ic multiplication and one-sided multiplication. The noncommutativity a
nd nonassociativity of these division algebras plays a crucial role in
our constructions.