" tommy quaternion "

[tommy1729]

The 8 dimensional " tommy octonion " - or whatever we will call it - is not constructed at random nor by trial and error.
It has to satisfy properties and those properties help in the construction.

For instance commutative implies (x*y)^2 = x^2 * y^2. So that is " respected " in the construction.
That also implies how many -1 and +1 we have in the table.

Another thing is (x*x)*y = x*(x*y). This also helps alot. ( alternative algebra , jordan algebra , power associative )

I also did a subalgebra test ( the analogue of a subgroup test ).

To be more specific we have dimension 8.

If dimension 8 has a subalgebra then it must be one of order 4 and of order 2 ( since 2*4 = 8 analogue to subgroups ).

So we only need to check for subgroups of order 2.

So I checked all the A,B,C,D for being the complex imaginary i^2 = -1 and all the E,F,G for being the split complex number j^2 = 1.

None of them matched so the split-complex or complex numbers are NOT a subalgebra.

Hence the " tommy quaternion " is not an extension.

For the subalgebra test see the 2 added pics.


regards

tommy1729