The quaternion conjugate has the property:
$$ \left( \boldsymbol{p} \otimes \boldsymbol{q} \right)^* = \boldsymbol{q}^* \otimes \boldsymbol{p}^* $$
The quaternion magnitude has the property: $$ \Vert\boldsymbol{p} \otimes \boldsymbol{q}\Vert = \Vert\boldsymbol{q} \otimes \boldsymbol{p} \Vert = \Vert\boldsymbol{p}\Vert \Vert\boldsymbol{q}\Vert $$
The quaternion product is not commutative: $$ \boldsymbol{p} \otimes \boldsymbol{q} \ne \boldsymbol{q} \otimes \boldsymbol{p} $$
The quaternion product is associative:
$$ \left(\boldsymbol{p} \otimes \boldsymbol{q}\right) \otimes \boldsymbol{r} = \boldsymbol{p} \otimes \left( \boldsymbol{q} \otimes \boldsymbol{r} \right) $$
The quaternion product is distributive: $$ \boldsymbol{p} \otimes \left( \boldsymbol{q} + \boldsymbol{r} \right) = \boldsymbol{p} \otimes \boldsymbol{q} + \boldsymbol{p} \otimes \boldsymbol{r} \qquad and \qquad \left( \boldsymbol{p} + \boldsymbol{q} \right) \otimes \boldsymbol{r} = \boldsymbol{p} \otimes \boldsymbol{r} + \boldsymbol{q} \otimes \boldsymbol{r} $$
The quaternion product of two quaternions is bi-linear, and may be expressed as two equivalent matrix products: $$ \boldsymbol{q}_1 \otimes \boldsymbol{q}_2 = [\boldsymbol{q_1}]_L \boldsymbol{q}_2 \qquad \boldsymbol{q}_1 \otimes \boldsymbol{q}_2 = [\boldsymbol{q_2}]_R \boldsymbol{q}_1 $$
Where $[q]_L$ and $[q]_R$ represent the left and right matrix products as described:
$$ [\boldsymbol{q}]_L = \begin{bmatrix} \phantom{-}q_0 & -q_1 & -q_2 & -q_3 \\ \phantom{-}q_1 & \phantom{-}q_0 & -q_3 & \phantom{-}q_2 \\ \phantom{-}q_2 & \phantom{-}q_3 & \phantom{-}q_0 & -q_1 \\ \phantom{-}q_3 & -q_2 & \phantom{-}q_1 & \phantom{-}q_0 \\ \end{bmatrix} = q_0\boldsymbol{I} + \begin{bmatrix} \phantom{-}0 & -\boldsymbol{q}^T_v \\ \phantom{-}\boldsymbol{q}_v & \phantom{-} \left[ \boldsymbol{q}_v \right]_{\times} \\ \end{bmatrix} $$
$$ [\boldsymbol{q}]_R = \begin{bmatrix} \phantom{-}q_0 & -q_1 & -q_2 & -q_3 \\ \phantom{-}q_1 & \phantom{-}q_0 & \phantom{-}q_3 & -q_2 \\ \phantom{-}q_2 & -q_3 & \phantom{-}q_0 & \phantom{-}q_1 \\ \phantom{-}q_3 & \phantom{-}q_2 & -q_1 & \phantom{-}q_0 \\ \end{bmatrix} = q_0\boldsymbol{I} + \begin{bmatrix} \phantom{-}0 & -\boldsymbol{q}^T_v \\ \phantom{-}\boldsymbol{q}_v & -[\boldsymbol{q}_v]_{\times} \\ \end{bmatrix} $$
Where $[\boldsymbol{q}_v]_{\times}$ represents the cross-product operation matrix, as depicted by $[a]_{\times}$:
$$ [\boldsymbol{a}]_{\times} = \begin{bmatrix} \phantom{-}0 & -a_3 & \phantom{-}a_2 \\ \phantom{-}a_3 & \phantom{-}0 & -a_1 \\ -a_2 & \phantom{-}a_1 & \phantom{-}0 \\ \end{bmatrix} $$
Therefore the associative property of the quaternions can be re-written:
$$ \left( \boldsymbol{q} \otimes \boldsymbol{x} \right) \otimes \boldsymbol{p} = [\boldsymbol{p}]_R [\boldsymbol{q}]_L \boldsymbol{x} \qquad and \qquad \boldsymbol{q} \otimes \left( \boldsymbol{x} \otimes \boldsymbol{p} \right) = [\boldsymbol{q}]_L [\boldsymbol{p}]_R\boldsymbol{x} $$
$$ [\boldsymbol{p}]_R[\boldsymbol{q}]_L = [\boldsymbol{q}]_L[\boldsymbol{p}]_R $$
The Quaternion Commutator is defined as: $$ \boldsymbol{p} \otimes \boldsymbol{q} - \boldsymbol{q} \otimes \boldsymbol{p} = 2 \boldsymbol{p}_v \times \boldsymbol{q}_v $$
The Product of pure quaternions, where $q_0$ is equal to zero and $\boldsymbol{q} = [0, \boldsymbol{q}_v]$, we have:
$$ \boldsymbol{p}_v \otimes \boldsymbol{q}_v = -\boldsymbol{p}^T_v\boldsymbol{q}_v + \boldsymbol{p}_v \times \boldsymbol{q}_v = \begin{bmatrix} -\boldsymbol{p}^T_v\boldsymbol{q}_v \\ \phantom{-}\boldsymbol{p}_v \times \boldsymbol{q}_v \\ \end{bmatrix} $$
$$ \boldsymbol{q}_v \otimes \boldsymbol{q}_v = -\boldsymbol{q}^T_v\boldsymbol{q}_v = -\Vert\boldsymbol{q}_v\Vert $$