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There are two fundamentally different forms octonion algebra can take.
They could be called Left Octonion Algebra and Right Octonion Algebra.
They are not isomorphic algebras.

However, this does not matter, since all meaningful physical results
derived from an octonion representation are algebraic invariants,
meaning the results will not care which of the 16 possible ways to
roll out the algebra is used.

Octonion algebra provides a suitable rank-1 hypercomplex basis for the
classical Electric and Magnetic field types based on their intrinsic
multiplication properties. When the algebra is used to describe the
Electrodynamic fields, there are additional rotational and
irrotational fields indicated. Perhaps the irrotational field that
rides along with the magnetic field in three of the algebraic units is
the gravitational field.

Using Electrodynamics as a roadmap, the form for an 8-current density
can be produced, and a suitable Action Function can be found in the
invariant portion of the Octonion product of field and 8-current. All
expected forms from Electrodynamics for work and force are found
within this Action Function.

The "Law of Algebraic Invariance" can be used to write the Action
Function in an integrable form. This identity can be thought of as the
divergence of the Octonion Stress-Energy-Momentum "tensor". Quotes
because the form is not your grandfather's tensor, it belongs to the
algebra of Octonions. Integration of this identity over the spatial 7-
volume produces the Octonion Conservation of Energy and Momentum.

The 8-potentials can be restricted to only expected functionals from
Electrodynamics. When this is done, the Octonion Stress-Energy-
Momentum "tensor" is identical to that of Electrodynamics, and its
divergence duplicates the work-force from Electrodynamics.

The derivation relies only on the full definition of the algebra, and
the belief that reality does not care which representation of the
algebra is used. Absolutely nothing else is inserted.

For more details, see

Rick