Quaternion Modulus in Terms of Conjugate

Theorem
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.

Let $\size {\mathbf x}$ be the modulus of $\mathbf x$.

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

Then:
 * $\size {\mathbf x}^2 \mathbf 1 = \mathbf x \overline {\mathbf x}$

Proof
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$.

Then: