Inversion of Double-Covering Map $\mathrm{Spin}(n) \rightarrow \mathrm{SO}(n, {\mathbb{R}})$ for $n\leq 6$

Abstract

This work provides an algorithmic procedure for finding the pair of elements in the spin group which map to a given matrix in the special orthogonal group of order five or six. This is achieved by first solving the problem when the special orthogonal matrix is a Givens rotation, and then exploiting the fact that the covering maps are group homomorphisms and that any special orthogonal matrix can be explicitly decomposed into a product of Givens rotations. For this purpose systems of quadratic equations in several variables have to be solved symbolically. The resulting solution display a transparent dependency on the entries of the Givens matrices.