Octonions
The Octonions, also called the Cayley-Dixon algebra, defined over a
commutative ring are an eight-dimensional non-associative algebra. Their
construction from quaternions is similar to the construction of quaternions
from complex numbers (see Quaternion).
As Octonion creates an eight-dimensional
algebra, you have to give eight components to construct an octonion.
Or you can use two quaternions to create an octonion.
You can easily demonstrate the non-associativity of multiplication.
As with the quaternions, we have a real part, the imaginary parts i, j,
k, and four additional imaginary parts E, I, J, and K. These parts
correspond to the canonical basis (1,i,j,k,E,I,J,K). For each basis
element there is a component operation to extract the coefficient of
the basis element for a given octonion.
A basis with respect to the quaternions is given by (1,E). However, you
might ask, what then are the commuting rules? To answer this, we create
some generic elements. We do this in Axim by simply changing the ground
ring from
Integer to
Polynomial Integer.
Note that quaternions are automatically converted to octonions in the
obvious way.
Finally, we check that the norm, defined as
the sum of the squares of the coefficients, is a multiplicative map.
Since the result is 0, the norm is multiplicative
Issue the system command
to display the list of operations defined by
Octonion.