Product of Quaternion Conjugates

Theorem
Let $\mathbf x, \mathbf y \in \mathbb H$ be quaternions.

Let $\overline{\mathbf x}$ be the conjugate of $\mathbf x$.

Then:
 * $\overline{\mathbf x \times \mathbf y} = \overline{\mathbf y} \times \overline{\mathbf x}$

but in general:
 * $\overline{\mathbf x \times \mathbf y} \ne \overline{\mathbf x} \times \overline{\mathbf y}$

Proof
Consider the matrix form of $\mathbf x$ and $\mathbf y$:


 * $\mathbf x = \begin{bmatrix} a & b \\ -\overline b & \overline a \end{bmatrix}$


 * $\mathbf y = \begin{bmatrix} c & d \\ -\overline d & \overline c \end{bmatrix}$

where $a, b, c, d \in \C$.

but: