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The Amitsur-Levitzki identity for matrices was generalized in severaldirections: by Kostant for simple finite-dimensional Lie algebras, by Kirillov(later joined by Kontsevich, Molev, Ovsienko, and Udalova) for simple vectorialLie algebras with polynomial coefficients, and by Gie, Pinczon, and Ushirobirafor the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|n)$. Dzhumadildaevswitched the focus of attention in these results by considering the algebraformed by antisymmetrizors and discovered a hidden supersymmetry ofcommutators. We overview these results and their possible generalizations (openproblems).

The Dzhumadildaev brackets: a hidden supersymmetry of commutators and the Amitsur-Levitzki-type identities