Freudenthal triple systems by root system methods

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, g_1, 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 = E_8 (where g_1 is the minuscule representation of E_7) and g = D_4, we determine the groups stabilizing the quartic form and both the quartic and bilinear forms.