Freudenthal triple systems by root system methods

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For certain Lie algebras g, we can use a Z/5Z-grading and define a quartic form and a skew-symmetric bilinear form on the degree 1 component, g₁, thereby constructing a Freudenthal triple system. The structure of the Freudenthal triple system is examined using root system methods available in the Lie algebra context. In the cases g = E₈ (where g₁ is the minuscule representation of E₇) and g = D₄, we determine the groups stabilizing the quartic form and both the quartic and bilinear forms.