  We give an explicit algebraic description of finite Lorentz transformations
of vectors in 10-dimensional Minkowski space by means of a parameterization in
terms of the octonions. The possible utility of these results for superstring
theory is mentioned. Along the way we describe automorphisms of the two highest
dimensional normed division algebras, namely the quaternions and the octonions,
in terms of conjugation maps. We use similar techniques 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
non-commutativity and non-associativity of these division algebras plays a
crucial role in our constructions.
