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 $\mathbf 1 \mathbf x$.