Every Albert division algebra over any field has an isotope that contains a cyclic cubic subfield, and its structure group then contains a subgroup of type ^3D4.
Hooda and Maneesh Thakur, Rational subgroups and invariants of F4, Israel Journal of Mathematics TBD (2019), 1-47
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The cyclicity problem for Albert algebras
Every Albert division algebra over any field has an isotope that contains a cyclic cubic subfield, and its structure group then contains a subgroup of type ^3D4.