I know the ordering of complex numbers -> quaternions -> octonions, so the octonions are basically built on top of the complex numbers. But I am wondering, still, if there is a way to reformulate them to not rely on the complex numbers, or to rework the complex numbers perhaps into simply ordered pairs of real numbers (so, vectors, rather than complex numbers), and so have sort of octonions but built out of primitives of vectors rather than complex numbers.


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