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other similar properties. It might also not be wise to use the same letter for a (pseudo) vector and a pseudoscalar, even if they have similar properties under multiplication. Perhaps we could instead refer to the unit pseudoscalar as k, to keep consistent with the notation used for split-quaternions? This is just a suggestion, though. Also, I am pretty sure that multiplication table at the bottom is inconsistent with itself. It seems that the intention is that j_x, j_y, j_z, i_x, i_y, i_z, and j are meant to correspond to li, lj, lk, i, j, k, and l from the table at the top of the page, respectively. Also, what are the intended domains for the objects undergoing a boost / rotation? I know little about the
Lorentz boost, but the rotation formula seems to only work for an "i" vector.
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I think the convention set by M. Gogberashvili and the rest of the Split-Octonian page is probably the better way to go, i.e. making j a polar vector and i an axial vector. In particular, this better agrees with the fact that the (cross) product of two perpendicular vectors is a pseudovector, among
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seems to deal only with specific numbers. The tag that this article is part of that project was added by a bot. Mistake? I doubt there are any specific split-octonions of sufficient note for its own article anyway--I doubt even quaternions have that distinction.
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What is not immediately obvious is the natural interpretation of the split-octonions this derivation brings with it. Three of the roots of unity are axial vectors and the three roots of -1 are polar vectors. The final root of unity represents a pseudo scalar.
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I will define the domain of x in the article. The rotation formula should rotate both a j and an i vector in a right handed fashion. This will not work with the first multiplication table, and is the primary motivation for the production of the second
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I was toying with the idea of starting with the split-complex numbers for a one dimensional space and using the Cayley-Dickson construction with a split root to add another dimension. Clearly this leads to the split-octonions.
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The sign convention of this table differs from the table above, and M. Gogberashvili, by the equivalent of flipping the direction of the circle in the Fano Plane. This is the source of the changes that you observed.
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As for the names i and j, the preference appears to be dependent on whether one considers the dot product or cross product more fundamental. The idea of naming the pseudoscalar k is compelling.
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Upon further review, the table was wrong. Flipping the direction of the inner circle alone is not a valid isomorphism of the split-octonions.
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The paper cited by M. Gogberashvili uses a similar notation, conveying the same idea but having a different sign convention.
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Specifically, the second table adopts the opposite sign for l, li, lj and lk given in the opening discussion.
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I deleted this tag. I would argue, however, that the quaternions i, j, and k deserve a page.
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