Hyperkähler manifolds from the Tits–Freudenthal magic square

We suggest a way to associate to each Lie algebra of type $G_2,D_4,F_4,E_6$, $E_7,E_8$ a family of polarized hyperkähler fourfolds, constructed as parametrizing certain families of cycles of hyperplane sections of certain homogeneous or quasi-homogeneous varieties. These cycles are modeled on the Legendrian varieties studied by Freudenthal in his geometric approach to the celebrated Tits–Freudenthal magic square of Lie algebras.