We have already seen that Geometric Algebra takes a vector algebra of dimension 'n' and generates an algebra of dimension n^{2}.

So the properties of the basis vectors largely determine the properties of the entire algebra. Unless otherwise stated the algebras will be based on conventional vectors which square to positive numbers, however it is possible to base the algebra on vectors which square to negative numbers or zero.

So we can define a geometric algebra by its signature of the form: G_{p,q,r}

where:

- p = number of basis vectors which square to a positive number
- q = number of basis vectors which square to a negative number
- r = number of basis vectors which square to zero

We can take subsets of these algebras, denoted as follows:

- G
^{+}= take only even grades of the algebra (0=scalar,2=bivector...) - G
^{-}= take only odd grades of the algebra (1=vector,3-trivector...)

Here are some examples:

Geometric Algebra based on | Designation |
---|---|

2D mulitvector based on standard vector geometry | G_{2,0,0} |

3D mulitvector based on standard vector geometry | G_{3,0,0} |

4D mulitvector based on vectors which square to positive | G_{4,0,0} |

complex | G_{0,1,0} |

dual | G_{0,0,1} |

double | G_{1,0,0} |

quaternions | G^{+}_{3,0,0} |

space-time algebra (STA) | G_{1,3,0} |

So can we generate a geometric algebra from any other algebra? Can we generate a geometric algebra from another geometric algebra?