Definition 22.1:

A pure quaternion is one whose real part is zero. Thus a quaternion is pure $q$ if and only if $\overline{q} = -q$, where the bar denotes the conjugate of $q$. We denote the set of all pure quaternions by $\Pi$. Thus $\Pi$ is a three dimensional real vector space with the Euclidean norm inherited from $\mathbb{R}^4$.

We now list three lemmas whose proofs are left for the reader as easy exercises in linear algebra. It is useful to recall that a linear map of $\mathbb{R}^n$ to itself which preserves the Euclidean norm is an orthogonal transformation.