Springer forms and the first Tits construction of exceptional Jordan division algebras (Q791624)

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.