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

is not 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 Field Norm of Quaternion is Positive Definite‎.

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

$N3$: Triangle Inequality
For example:


 * $\map n {1 + 1} = 4 > 2 = \map n 1 + \map n 1$

and so $N3$ is not satisfied.

Not all the norm axioms are fulfilled.

Hence the result.