## Prerequisites

If you are not familiar with this subject you may like to look at the following pages first:

## Description

We could either start with a quaternion and make each element of the quaternion a double. This means that there are now 8 elements, the 4 quaternion elements (real, i, j and k) and their doubles (D, iD, jD and kD).

So, given the Cayley Table for Double:

a*b | b.1 | b.D |

a.1 | 1 | D |

a.D | D | 1 |

and the Cayley Table for quaternions:

a*b |
b.1 | b.i | b.j | b.k |

a.1 | 1 | i | j | k |

a.i | i | -1 | k | -j |

a.j | j | -k | -1 | i |

a.k | k | j | -i | -1 |

we want to combine them to give an 8x8 Cayley Table as follows:

### Table for: Double Quaternion

a*b | b.1 | b.i | b.j | b.k | b.D | b.Di | b.Dj | b.Dk |

a.1 | 1 | i | j | k | D | Di | Dj | Dk |

a.i | i | -1 | k | -j | Di | -D | -Dk | Dj |

a.j | j | -k | -1 | i | Dj | Dk | -D | -Di |

a.k | k | j | -i | -1 | Dk | -Dj | Di | -D |

a.D | D | -Di | -Dj | -Dk | 1 | -i | -j | -k |

a.Di | Di | D | -Dk | Dj | i | 1 | k | -j |

a.Dj | Dj | Dk | D | -Di | j | -k | 1 | i |

a.Dk | Dk | -Dj | Di | D | k | j | -i | 1 |

analysing commutivity: table does not commute: for example: i*j != j*i

analysing associativity: table does not associate, for example, (i* j)* D=k* D=Dk is not equal to i*(j* D)=i*Dj=-Dk

how these results were generated.

As the above link explains, the table was generated by a computer program using a modification of Cayley-Dickson process.

### Inverse of double quaternion

Dividing quaternions is done by multiplying top and bottom by the conjugate, can we do the same with double quaternions?

How do we define the conjugate of dual quaternions?