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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.

Hyperkähler manifolds from the Tits–Freudenthal magic square