Freudenthal triple systems by root system methods

For certain Lie algebras g, we can use a grading g=g−2⊕g−1⊕g0⊕g1⊕g2 and define a quartic form and a skew-symmetric bilinear form on 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.