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The odd-ordered subalgebra (x, y, z, xyz) is symmetrical to the even-ordered subalgebra (1, xy, yz, zx), and can also represent quaternions.

The odd-ordered subspace is a subspace, not a subalgebra, because multiplication is not closed.

That doesn't work, the odd space isn't closed.

xx = 1

I don't know what I was thinking....