Cubic subfields of exceptional simple Jordan algebras

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Let E / k E/k be a cubic field extension and J J a simple exceptional Jordan algebra of degree 3 over k k . Then E E is a reducing field of J J if and only if E E is isomorphic to a (maximal) subfield of some isotope of J J . If k k has characteristic not 2 or 3 and contains the third roots of unity then every simple exceptional Jordan division algebra of degree 3 over k k contains a cyclic cubic subfield.