Maths - Octonion and Clifford Algebra - Code to generate the tables

Here is how I generated the tables for this page.

The tables were generated using this program.

To produce the results the program needs to have an XML input code. Here I have listed this input code next to the output of the program:

 

code program output
<classDef>
<outputTable type="product" format="html" name="table" analyse="on">
<varDef name="a" type="oct" sign="0" subAlgebra="all"/>
</outputTable>
</classDef>

Octonion

This is the table often given for octonions:
e e1 e2 e3 e4 e5 e6 e7
e1 -e e4 e7 -e2 e6 -e5 -e3
e2 -e4 -e e5 e1 -e3 e7 -e6
e3 -e7 -e5 -e e6 e2 -e4 e1
e4 e2 -e1 -e6 -e e7 e3 -e5
e5 -e6 e3 -e2 -e7 -e e1 e4
e6 e5 -e7 e4 -e3 -e1 -e e2
e7 e6 e6 -e1 e5 -e3 -e2 -e

analysing commutivity: table does not commute:
for example: e1*e2 != e2*e1

analysing associativity: table does not associate,
for example, (e1* e2)* e3=e4* e3=-e6
is not equal to e1*(e2* e3)=e1*e5=e6
<classDef>
<outputTable type="product" format="html" name="table" analyse="on">
<varDef name="a" type="oct2" sign="0" subAlgebra="all"/>
</outputTable>
</classDef>

Octonion (rearranged)

The table can be rearranged to give a table which looks closer to the Clifford algebras for easier comparison:
e e1 e2 e3 e4 e5 e6 e7
e1 -e -e3 e2 -e5 e4 e7 -e6
e2 e3 -e -e1 -e6 -e7 e4 e5
e3 -e2 e1 -e -e7 e6 -e5 e4
e4 e5 e6 e7 -e -e1 -e2 -e3
e5 -e4 e7 -e6 e1 -e e3 -e2
e6 -e7 -e4 e5 e2 -e3 -e e1
e7 e6 -e5 -e4 e3 e2 -e1 -e

analysing commutivity: table does not commute:

for example: e1*e2 != e2*e1

analysing associativity:

table does not associate,
for example, (e1* e2)* e4=-e3* e4=e7
is not equal to e1*(e2* e4)=e1*-e6=-e7

<classDef>
<outputTable type="product" format="html" name="table" analyse="on">
<mathTypeMulti name="a" type="4" sign="0" subAlgebra="even"/>
</outputTable>
</classDef>

Multivector 4D Even

table
e e12 e31 e23 e41 e42 e43 e1234
e12 -e e23 -e31 -e42 e41 -e1234 e43
e31 -e23 -e e12 e43 -e1234 -e41 e42
e23 e31 -e12 -e -e1234 -e43 e42 e41
e41 e42 -e43 -e1234 -e -e12 e31 e23
e42 -e41 -e1234 e43 e12 -e -e23 e31
e43 -e1234 e41 -e42 -e31 e23 -e e12
e1234 e43 e42 e41 e23 e31 e12 e

analysing commutivity:

table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates
<classDef>
<outputTable type="product" format="html" name="table" analyse="on">
<mathTypeMulti name="a" type="4" sign="1" subAlgebra="even"/>
</outputTable>
</classDef>

Multivector 4D Even(1 dimension squares to -ve)

table
e e12 e31 e23 e41 e42 e43 e1234
e12 e -e23 -e31 e42 e41 -e1234 -e43
e31 e23 e e12 -e43 -e1234 -e41 -e42
e23 e31 -e12 -e -e1234 -e43 e42 e41
e41 -e42 e43 -e1234 e -e12 e31 -e23
e42 -e41 -e1234 e43 e12 -e -e23 e31
e43 -e1234 e41 -e42 -e31 e23 -e e12
e1234 -e43 -e42 e41 -e23 e31 e12 -e
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1 analysing associativity: table associates
<classDef>
<outputTable type="product" format="html" name="table" analyse="on">
<mathTypeMulti name="a" type="4" sign="3" subAlgebra="even"/>
</outputTable>
</classDef>

Multivector 4D Even(2 dimensions squares to -ve)

table:
e e12 e31 e23 e41 e42 e43 e1234
e12 -e -e23 e31 e42 -e41 -e1234 e43
e31 e23 e e12 -e43 -e1234 -e41 -e42
e23 -e31 -e12 e -e1234 e43 e42 -e41
e41 -e42 e43 -e1234 e -e12 e31 -e23
e42 e41 -e1234 -e43 e12 e -e23 -e31
e43 -e1234 e41 -e42 -e31 e23 -e e12
e1234 e43 -e42 -e41 -e23 -e31 e12 e
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1 analysing associativity: table associates
<classDef>
<outputTable type="product" format="html" name="table" analyse="on">
<mathTypeMulti name="a" type="4" sign="7" subAlgebra="even"/>
</outputTable>
</classDef>

Multivector 4D Even(3 dimensions squares to -ve)

table
e e12 e31 e23 e41 e42 e43 e1234
e12 -e -e23 e31 e42 -e41 -e1234 e43
e31 e23 -e -e12 -e43 -e1234 e41 e42
e23 -e31 e12 -e -e1234 e43 -e42 e41
e41 -e42 e43 -e1234 e -e12 e31 -e23
e42 e41 -e1234 -e43 e12 e -e23 -e31
e43 -e1234 -e41 e42 -e31 e23 e -e12
e1234 e43 e42 e41 -e23 -e31 -e12 -e
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1 analysing associativity: table associates
<classDef>
<outputTable type="product" format="html" name="table" analyse="on">
<mathTypeMulti name="a" type="4" sign="15" subAlgebra="even"/>
</outputTable>
</classDef>

Multivector 4D Even(4 dimension squares to -ve)

table
e e12 e31 e23 e41 e42 e43 e1234
e12 -e -e23 e31 e42 -e41 -e1234 e43
e31 e23 -e -e12 -e43 -e1234 e41 e42
e23 -e31 e12 -e -e1234 e43 -e42 e41
e41 -e42 e43 -e1234 -e e12 -e31 e23
e42 e41 -e1234 -e43 -e12 -e e23 e31
e43 -e1234 -e41 e42 e31 -e23 -e e12
e1234 e43 e42 e41 e23 e31 e12 e
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1 analysing associativity: table associates.

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