Some New Results on Three-Dimensional Rotations and Pseudo-Rotations

We use a vector parameter technique to obtain the generalized Euler decompositions with respect to arbitrarily chosen axes for the three-dimensional special orthogonal group SO(3) and the three-dimensional Lorentz group SO(2,1). Our approach, based on projecting a quaternion (respectively split quaternion) from the corresponding spin cover, has proven quite effective in various problems of geometry and physics [1, 2, 3]. In particular, we obtain explicit (generally double-valued) expressions for the three parameters in the decomposition and discuss separately the degenerate and divergent solutions, as well as decompositions with respect to two axes. There are some straightforward applications of this method in special relativity and quantum mechanics which are discussed elsewhere (see [4]).