Non-associative Frobenius algebras of type [formula omitted] with trivial Tits algebras

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Very recently, Maurice Chayet and Skip Garibaldi have introduced a class of commutative non-associative algebras, for each simple linear algebraic group over an arbitrary field (with some minor restriction on the characteristic). In a previous paper, we gave an explicit description of these algebras for groups of type G2 and F4 in terms of the octonion algebras and the Albert algebras, respectively. In this paper, we attempt a similar approach for type E6.