We use the mathematical notation of adorno2017 unless otherwise stated.
Consider these important definitions that apply to all following explanations.
The quaternion set is given by
$\mathbb{H}\triangleq\left\{ h_{1}+\imi h_{2}+\imj h_{3}+\imk h_{4}\,:\,h_{1},h_{2},h_{3},h_{4}\in\mathbb{R}\right\}$
in which the imaginary units $\imi$, $\imj$, and $\imk$ have the following properties:
$\hat{\imath}^{2}=\hat{\jmath}^{2}=\hat{k}^{2}=\hat{\imath}\hat{\jmath}\hat{k}=-1$
The dual quaternion set is given by
$\mathcal{H}\triangleq\left\{ \quat h+\dual\quat h'\,:\,\quat h,\quat h'\in\mathbb{H},\,\dual^{2}=0,\,\dual\neq0\right\}$
where $\dual^2=0$ but $\dual\neq0$.