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:

\(\ds \mathbf x \mathbf 1\)

\(=\)

\(\ds \paren {a \cdot 1 - b \cdot 0 - c \cdot 0 - d \cdot 0} \mathbf 1\)

\(\ds \)

\(+\)

\(\ds \paren {a \cdot 0 + b \cdot 1 + c \cdot 0 - d \cdot 0} \mathbf i\)

\(\ds \)

\(+\)

\(\ds \paren {a \cdot 0 - b \cdot 0 + c \cdot 1 + d \cdot 0} \mathbf j\)

\(\ds \)

\(+\)

\(\ds \paren {a \cdot 0 + b \cdot 0 - c \cdot 0 + d \cdot 1} \mathbf k\)

\(\ds \)

\(=\)

\(\ds a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k\)

\(\ds \)

\(=\)

\(\ds \mathbf x\)

Similarly for $\mathbf 1 \mathbf x$.

$\blacksquare$