  We suggest a way to associate to each Lie algebra of type G2, D4, F4, E6, E7,
E8 a family of polarized hyperkahler 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.
