Tensor products of composition algebras, Albert forms and some exceptional simple Lie algebras

Allison, B. N.

doi:10.1090/S0002-9947-1988-0933312-2

Tensor products of composition algebras, Albert forms and some exceptional simple Lie algebras
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Trans. Amer. Math. Soc. 306 (1988), 667-695 Request permission

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 ${\mathbf {R}}(({T_1}, \ldots ,{T_n})),\;n \leqslant 3$.

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