Inversion of double-covering map \(SPIN(N) \to SO (N, {\mathbb R})\) for \(N \leq 6\) (Q526073)

For any integer \(n\), the spin group \(\text{Spin}(n)\) is the double cover of the special orthogonal group \(\text{SO}(n,\mathbb{R})\) such that there is a short exact sequence of Lie groups \[ 1 \rightarrow \mathbb{Z}/2 \mathbb{Z} \rightarrow \text{Spin}(n) \rightarrow \text{SO}(n,\mathbb{R}) \rightarrow 1. \] The cases \(n=3\) and \(4\), where \(\text{Spin}(n)=\text{SU}(2)\) and \(\text{SU}(2) \times \text{SU}(2)\) respectively, are classical and date back to Hamilton, who described both the covering maps and the co-images using the quaternion algebra \(\mathbb{H}\). In the paper under discussion, the authors consider the cases \(n=5\) and 6, where \(\text{Spin}(n)=\text{Sp}(4)\) and \(\text{SU}(4)\) respectively. They provide explicit formulas for the covering maps, and algorithms for finding the co-images.