The octonion (Cayley) algebra is studied in a split basis by means of a formalism that brings outs its quark structure. The groups SO(8), SO(7), and G2 are represented by octonions as well as by 8 × 8 matrices and the principle of triality is studied in this formalism. Reduction is made through the physically important subgroups SU(3) and SU(2) ⊗ SU(2) of G2, the automorphism group of octonions.

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27.
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28.
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29.
Note that we put a bar over the indexed matrices when they act on the split octonions, i.e., under the numbering: (u1u2u3u0u1*u2*u3*μ0*)(s1s2s3s4s5s6s7s8) we have Ēabsc = δbcsa,a,b,c = 1,…,8. For the real octonions we have the numbering (eA,1)↔(eA,e8),A = 1,…,7. Then Eabec = δbcea,a,b,c, = 1,…,8. We also defined ΣabΣab = Eab−Eba,Σab = Eab−Eba.
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For a proof and more details see N. Jacobson Lie Algebras (Interscience, New York, 1962.).
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