Multiplicative Identity for Quaternions

Theorem
In the set of quaternions $$\mathbb H$$, the element:
 * $$\mathbf 1 + 0 \mathbf i + 0 \mathbf j + 0 \mathbf k$$

serves as the identity element for quaternion multiplication.

This element is written $$\mathbf 1$$.

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

From the definition of quaternion multiplication:

$$ $$ $$ $$ $$ $$

Similarly for $$\mathbf 1 \mathbf x$$.