Multiplicative Identity for Quaternions

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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:

\(\displaystyle \mathbf x \mathbf 1\) \(=\) \(\displaystyle \left({a \cdot 1 - b \cdot 0 - c \cdot 0 - d \cdot 0}\right) \mathbf 1\)
\(\displaystyle \) \(+\) \(\displaystyle \left({a \cdot 0 + b \cdot 1 + c \cdot 0 - d \cdot 0}\right) \mathbf i\)
\(\displaystyle \) \(+\) \(\displaystyle \left({a \cdot 0 - b \cdot 0 + c \cdot 1 + d \cdot 0}\right) \mathbf j\)
\(\displaystyle \) \(+\) \(\displaystyle \left({a \cdot 0 + b \cdot 0 - c \cdot 0 + d \cdot 1}\right) \mathbf k\)
\(\displaystyle \) \(=\) \(\displaystyle a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k\)
\(\displaystyle \) \(=\) \(\displaystyle \mathbf x\)

Similarly for $\mathbf 1 \mathbf x$.

$\blacksquare$