# Definition:Quaternion/Mistake

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## Contents

## Source Work

1974: Robert Gilmore: *Lie Groups, Lie Algebras and Some of their Applications*:

- Chapter $1$: Introductory Concepts
- $1$. Basic Building Blocks

## Mistake

*Every quaternion can be represented in the form*- $q = q_0 1 + q_1 \lambda_1 + q_2 \lambda_2 + q_3 \lambda_3$

- where the $q_i \, \paren {i = 0, 1, 2, 3}$ are real numbers and the $\lambda_1$ have multiplicative properties defined by

\(\displaystyle \lambda_0 \lambda_i\) | \(=\) | \(\displaystyle \lambda_i \lambda_0 = \lambda_i\) | \(\displaystyle i = 0, 1, 2, 3\) | ||||||||||

\(\displaystyle \lambda_i \lambda_i\) | \(=\) | \(\displaystyle -\lambda_0\) | |||||||||||

\(\displaystyle \lambda_1 \lambda_2\) | \(=\) | \(\displaystyle -\lambda_2 \lambda_1 = \lambda_3\) | |||||||||||

\(\displaystyle \lambda_2 \lambda_3\) | \(=\) | \(\displaystyle -\lambda_3 \lambda_2 = \lambda_1\) | |||||||||||

\(\displaystyle \lambda_3 \lambda_1\) | \(=\) | \(\displaystyle -\lambda_1 \lambda_3 = \lambda_2\) |

## Correction

The representation of $q$ should read:

- $q = q_0 \lambda_0 + q_1 \lambda_1 + q_2 \lambda_2 + q_3 \lambda_3$

and the line:

- $\lambda_i \lambda_i = -\lambda_0$

holds only for $i = 1, 2, 3$ because:

- $\lambda_0 \lambda_0 = \lambda_0$

One suspects that the author conflated this presentation with one where $\lambda_0$ has been identified with the number $1$, and only partially has this been translated into its current form.

## Sources

- 1974: Robert Gilmore:
*Lie Groups, Lie Algebras and Some of their Applications*... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $3$. FIELD