The notion of commutator width of a group, defined as the smallest number of commutators needed to represent each element of the derived group as their product, has been extensively studied over the past decades. In particular, in [Math. Ann. 294, No. 2, 235--265 (1992; Zbl 0894.55006)] \textit{J. Barge} and \textit{E. Ghys} discovered the first example of a simple group of commutator width greater than one among groups of diffeomorphisms of smooth manifolds. \par We consider a parallel notion of bracket width of a Lie algebra and present the first examples of simple Lie algebras of bracket width greater than one. They are found among the algebras of algebraic vector fields on smooth affine varieties.

14H52, 17B66

simple Lie algebras, Lie algebras of algebraic, symplectic and Hamiltonian vector fields, smooth affine curves, Danielewski surfaces, locally nilpotent derivations

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IMB UMR5584, CNRS Université Bourgogne Franche-Comté, F-21000 Dijon, France

Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, Israel

Institut für Mathematik, Friedrich-Schiller-Universität Jena, Jena 07737, Germany