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Springer forms and the first Tits construction of exceptional Jordan division algebras. (English) Zbl 0536.17009
Let J be an absolutely simple Jordan division algebra of degree three over a field k of arbitrary characteristic. Let E be a separable cubic subfield of J. Necessary and sufficient conditions are given for J to contain a subalgebra constructed from E by the first Tits construction. These conditions depend on a quadratic form associated with E which was introduced by T. A. Springer. As a consequence, if k has characteristic $$\neq 3$$ and contains third roots of units, then a central exceptional Jordan division algebra J over k arises from the first Tits construction if and only if every reducing field of J also splits J. If J is an exceptional simple Jordan algebra arising from the first Tits construction, then every isotope of J is isomorphic to J.
Reviewer: R.Bix

##### MSC:
 17C10 Structure theory for Jordan algebras 17C20 Simple, semisimple Jordan algebras 17C40 Exceptional Jordan structures
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##### References:
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