Tensor Products of Composition Algebras, Albert Forms and Some Exceptional Simple Lie Algebras

Abstract

In this paper, we study algebras with involution that are isomorphic after base field extension to the tensor product of two composition algebras. To any such algebra $(\mathcal {A}, - )$, we associate a quadratic form $Q$ called the Albert form of $(\mathcal {A}, - )$. The Albert form is used to give necessary and sufficient conditions for two such algebras to be isotopic. Using a Lie algebra construction of Kantor, we are then able to give a description of the isomorphism classes of Lie algebras of index $F_{4,1}^{21}$, ${}^2E_{6,1}^{29}$, $E_{7,1}^{48}$ and $E_{8,1}^{91}$. That description is used to obtain a classification of the indicated Lie algebras over ${\mathbf {R}}(({T_1}, \ldots ,{T_n})),\;n \leqslant 3$.