The (𝐴₂,𝐺₂) duality in 𝐸₆, octonions and the triality principle

Rubenthaler, Hubert

doi:10.1090/S0002-9947-07-04269-9

The $(\textbf {A_2,G_2})$ duality in $\textbf {E_6}$, octonions and the triality principle
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Trans. Amer. Math. Soc. 360 (2008), 347-367 Request permission

Abstract:

We show that the existence of a dual pair of type $(A_2, G_{2})$ in $E_6$ leads to a definition of the product of octonions on a specific $8$-dimensional subspace of $E_6$. This product is expressed only in terms of the Lie bracket of $E_6$. The well known triality principle becomes an easy consequence of this definition, and $G_2$ acting by the adjoint action is shown to be the algebra of derivations of the octonions. The real octonions are obtained from two specific real forms of $E_6$.

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