The $({\bf A_2,G_2})$ duality in ${\bf E_6}$, octonions and the triality principle

Abstract

We show that the existence of an $A_2$--type subalgebra of $E_6$ which is dual to another one of $G_2$--type leads to a definition of the product of octonions on an $8$-dimensional subspace of $E_6$. Thisproduct is then only expressed in terms of the Lie bracket of $E_6$. The well knowntriality principle becomes an easy consequence of this definition and $G_2$ acting by the adjoint action isshown to be the algebra of derivations of the octonions. The real octonions are obtained from two specific real forms of $E_6$.