  The Amitsur-Levitzki identity for matrices was generalized in several
directions: by Kostant for simple finite-dimensional Lie algebras, by Kirillov
(later joined by Kontsevich, Molev, Ovsienko, and Udalova) for simple vectorial
Lie algebras with polynomial coefficients, and by Gie, Pinczon, and Ushirobira
for the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|n)$. Dzhumadildaev
switched the focus of attention in these results by considering the algebra
formed by antisymmetrizors and discovered a hidden supersymmetry of
commutators. We overview these results and their possible generalizations (open
problems).
