Product of Sums of Four Squares/Proof 2

Proof
Let:
 * $\mathbf m = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$
 * $\mathbf n = -w \mathbf 1 + x \mathbf i + y \mathbf j + z \mathbf k$

be two quaternions.

Then: