In order to work with rotations using the multiplication operator, in particular to combine two individual rotations to give an equivalent single rotation, then we need to use the even sub-algebra.

## Complex Numbers

So if we start with 2D euclidean space, that is two dimensions that both square to positive, then we get the Clifford algebra G 2,0,0:

e = scalar (squares to positive)

e_{1} = vector (squares to positive)

e_{2} = vector (squares to positive)

e_{12} = bivector (squares to negative)

Taking the even sub-algebra we take just the scalar and bivector dimensions:

e = scalar (squares to positive)

e_{12} = bivector (squares to negative)

This is equivalent (isomorphic to) complex number algebra.

## Double Numbers

This time we start with a vector space where one dimension squares to positive and the other squares to negative, then we get the Clifford algebra G 1,1,0:

e = scalar (squares to positive)

e_{1} = vector (squares to positive)

e_{2} = vector (squares to negative)

e_{12} = bivector (squares to positive)

Taking the even sub-algebra we take just the scalar and bivector dimensions:

e = scalar (squares to positive)

e_{12} = bivector (squares to positive)

This is equivalent (isomorphic to) double number algebra.

## Duality

So, from the above, there appears to be a duality between Euclidean and Minkowski space, where rotations in Euclidean space are calculated using Minkowski space and visa versa.