, Volume 45, Issue 3, pp 249-272

Springer forms and the first Tits construction of exceptional Jordan division algebras

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Abstract

In this paper, a certain quadratic form, originally due to Springer [15], which may be associated with any separable cubic subfield living inside an exceptional simple Jordan algebra is related to the coordinate algebra of an appropriate scalar extension. We use this relation to show that, in the presence of the third roots of unity, exceptional Jordan division algebras arising from the first Tits construction are precisely those where reducing fields and splitting fields agree, and that all isotopes of a first construction exceptional division algebra are isomorphic.

The research of the second author was supported in part by an NSERC grant and by a von Humboldt Fellowship at the University of Münster