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 of G2, the automorphism group of octonions.
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Note the distinction between the terms nonassociative and not associative. The former is generally used to denote all the composition algebras mentioned above which satisfy the property of alternativity defined below.
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H. Freudenthal, “Oktaven, Ausnahmegruppen and Oktavengeometrie,” (mimeographed), Utretch (1951).
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By three “independent” elements we mean any three elements such that none of them is proportional to a product of the other two, i.e.,
24.
An element left invariant by the Lie group is said to be annihilated by the Lie algebra.
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For example, take as a basis generating a quaternion algebra H. Then each octonion can be written as ,, where , with the product defined by (The bar denotes quaternion conjugation).
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 extends subgroup into U(3) and the group into
29.
Note that we put a bar over the indexed matrices when they act on the split octonions, i.e., under the numbering: we have For the real octonions we have the numbering Then We also defined
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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.
34.
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1973
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