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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26.
For example, take {1,e1,e2,e3} as a basis generating a quaternion algebra H. Then each octonion can be written as
z = q1+q2e7
,
w = r1+r2e7εO
, where
q1,q2,r1,r2εH
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27.
An equivalent form of this basis was first studied by G. Seligman as the derivation algebra of Zorn’s vector matrices given in Appendix B. See Ref. 17.
28.
In fact, the generator N3 extends SU(3) subgroup into U(3) and the group G2 into SO(7).
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.).
33.
Here we use capital letters for the elements of the Lie algebra and small letters for the group elements.
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See
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