Lemma 22.2:

If $q$ is a pure quaternion then so is $x^{-1}qx$ for any non-zero quaternion $x$.

Thus, each non-zero quaternion $x$ defines a non-singular linear map $T_{x}:\Pi\longrightarrow \Pi$ namely

$$T_{x} (q) = x^{-1}qx \eqno(22.1)$$