The basis vectors can be represented by matrices, this algebra was worked out independently by Pauli for his work on quantum mechanics. Murray GellMann defined an extention of Pauli matricies to 3x3 matricies:
The scalar would be the identity matrix.
The structure constant is antisymmetric in the three indicies and has values:
f^{123} = 2
f^{147} = f^{165} = f^{246} = f^{257} = f^{345} = f^{376} = 1
f^{458} = f^{678} =√ 3
The bivectors can be calculated by multiplying the matrices:
The trivectors are:
So the complete geometric multiplication table is:
a*b 
b.e 
b.e1 
b.e2 
b.e3 
b.e12 
b.e31 
b.e23 
b.e123 
a.e 




e_{12} 
e_{31} 
e_{23} 
e_{123} 
a.e1 


e_{12} 
e_{31} 


e_{123} 
e_{23} 
a.e2 

e_{12} 

e_{23} 

e_{123} 

e_{31} 
a.e3 

e_{31} 
e_{23} 

e_{123} 


e_{12} 
a.e12 
e_{12} 


e_{123} 

e_{23} 
e_{31} 

a.e31 
e_{31} 

e_{123} 

e_{23} 

e_{12} 

a.e23 
e_{23} 
e_{123} 


e_{31} 
e_{12} 


a.e123 
e_{123} 
e_{23} 
e_{31} 
e_{12} 




which is equivalent to the table derived here.
Further Reading
Other uses of Pauli Matrix:
Related Concepts:
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