On this page we will generate various 4 dimensional Cayley tables using combinations of the following methods:
 From complex and double numbers combined using the (modified) CayleyDickson method.
 From Clifford algebras based on 2D vector algebra (each of which may square to +ve or ve).
 From even subalgebras of 8D clifford algebras (again using various combinations of +ve and ve squaring basis).
The aim is to look for correspondences between the various types of algebras. We will see that, for these 4D algebras, that the CayleyDickson method produces the same algebras as the Clifford algebras:
 DD = G 2,0,0
 DC = G 1,1,0
 CD = G 1,1,0
 CC = G 0,2,0
When we double up to 8D algebras we will see that the algebras will start to diverge.
For two algebras to be isomorphic (that is there is a mapping between equivalent algebras) then the tables don't necessarily need to be identical, we can swap or invert any dimensions (see this page), so there are other equivalences that we can find. For instance:
DAs we have seen on this page the type of each entry will be common for all these methods, it will be:

or equivalently in quaternion notation: 

So to make the comparison clearer on the page we will only show the sign and colour code the entries so that the pattern will show:
4D CayleyDickson Algebras
DD 
DC  CD  CC  

i & j anticommute i*j = j*i 





i & j commute i*j = j*i 





i & j anticommute but left handed 




how these results were generated.
As the above link explains, the table was generated by a computer program from the (modified) CaleyDickson doubling process.
These results are labeled as follows:
 DD = Double where each element is a Double
 DC = Double where each element is a Complex
 CD = Complex where each element is a Double
 CC = Complex where each element is a Complex
Note that in this contextdenotes the direct product as explained on this page (Its not the kronecker product of the tables).
4D Clifford Algebras
That is, Clifford algebras based on 2 vector dimensions, I have tried the combinations of these dimensions squaring to positive and negative.
both dimensions square to +ve G 2,0,0 
one dimension squares to ve other to +ve G 1,1,0 
one dimension squares to +ve other to ve G 1,1,0 
both dimensions square to ve G 0,2,0 


vectors anticommute 




how these results were generated.
As the above link explains, the table was generated by a computer program from the ordering of bases.
Even Subalgebras of 8D Clifford Algebras
That is, An even subalgebra of Clifford algebras based on 3 vector dimensions.
G+ 3,0,0 
G+ 2,1,0 
G+ 1,2,0 
G+ 0,3,0 


vectors anticommute 



