Field Norm of Quaternion is Positive Definite

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

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

The field norm of $\mathbf x$:
 * $\map n {\mathbf x} := \cmod {\mathbf x \overline {\mathbf x} }$

is positive definite.

Proof
Hence $n$ is positive definite by definition.