Lemma 22.3:

The linear map $T_{x} : \mathbb{R}^4\longrightarrow \mathbb{R}^4$ given by equation (22.1) preserves the Euclidean norm and so defines an element of $O(4, \mathbb{R})$. Its restriction to $\Pi$ still denoted by $T_{x}$ maps $\Pi$ onto itself and so may be identified as an element of $O(3, \mathbb{R})$. The map

$$\psi: x\mapsto T_{x} \eqno(22.2)$$

is a group homomorphism from the multiplicative group of non-zero quaternions into $O(4, \mathbb{R})$.