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"Alas, the group SU(3) is not the unit octonions. The unit octonions do not form a group since they aren't associative. SU(3) is related to the octonions more indirectly. The group of symmetries (or technically, "automorphisms") of the octonions is the exceptional group G2, which contains SU(3). To get SU(3), we can take the subgroup of G2 that preserves a given unit imaginary octonion... say e1. This is how Dixon relates SU(3) to the octonions. However, why should one unit imaginary octonion be different from the rest? Some sort of "symmetry breaking", presumably? It seems a bit ad hoc."

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The complexified octonions (which I label S = C⊗O) is not a division algebra. Its identity can be resolved into a pair of orthogonal idempotents, and it is from this resolution of the identity that much of what's interesting and beautiful about S arises. This resolution requires a direction in imaginary O space be chosen, and the subgroup of G2 leaving this direction invariant is SU(3). With respect to this resolution S splits into 4 SU(3) multiplets: singlet; antisinglet; triplet; antitriplet. This notion was used by Gürsey at Yale in the 1970s. There's nothing ad hoc about it. It's an ineluctable and beautiful part of the maths. There's nothing vague about it, as two books and numerous papers have attempted to demonstrate ad nauseam. Sigh.

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# B Why SU(3)xSU(2)xU(1)?

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