Finite Lorentz transformations, automorphisms, and division algebras

An explicit algebraic description of finite Lorentz transformations of vectors in ten‐dimensional Minkowski space is given by means of a parametrization in terms of the octonions. The possible utility of these results for superstring theory is mentioned. Along the way automorphisms of the two highest dimensional normed division algebras, namely, the quaternions and the octonions, are described in terms of conjugation maps. Similar techniques are used to define SO(3) and SO(7) via conjugation, SO(4) via symmetric multiplication, and SO(8) via both symmetric multiplication and one‐sided multiplication. The noncommutativity and nonassociativity of these division algebras plays a crucial role in our constructions.