Lemma 22.4:

The kernel of $\psi$ is the set of non-zero real numbers. In particular the kernel of the map

$$\psi: S^3 \longrightarrow O(3, \mathbb{R}) \eqno(22.3)$$

obtained by restricting $\psi$ to $S^3$ is the two element group $\{\pm 1\}$.