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.