Field Norm of Quaternion is not Norm

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 norm of $\mathbf x$:
 * $n \left({\mathbf x}\right) := \left\vert{\mathbf x \overline {\mathbf x} }\right\vert$

is a norm in the abstract algebraic context of a division ring.

Proof
Each of the norm axioms is examined in turn:

$N1$: Positive Definiteness
This is proved in Norm of Quaternion is Positive Definite‎.

$N2$: Multiplicativity
This is proved in Norm of Quaternion is Multiplicative.

$N3$: Triangle Inequality
All the norm axioms are fulfilled, and the result follows.