Generalized Euler decompositions of some six-dimensional Lie groups

Abstract

Here we present the generalized Euler decompositions of the six-dimensional Lie groups SO(4), SO∗(4) and SO(2,2) using their (local) direct product structure [1] and a technique we have developed for SO(3) and SO(2,1) (cf. [2]). Although in even dimensions the Euler invariant axis theorem is not valid, one may introduce the notion of bi-axis n ⊗ ñ and decompose the generalized vector-parameter c ⊗ c with respect to a given set of bi-axes. As for the Lorentz group SO(3,1), we deal with complex vector-parameters [4] and the decomposition intertwines real and imaginary parts of vectors. Thus, bi-axes in that case have the interpretation of projective lines in parameter space CP3.