Bracket width of simple Lie algebras

For an element of a group G representable as a product of commutators, one can define its commutator length as the smallest number of commutators needed for such a representation, by definition the other elements are of infinite length. The commutator width of G is defined as supremum of the lengths of its elements. Recently it was proven that all finite simple groups have commutator width one. On the other hand, there are examples of infinite simple groups of arbitrary finite width and of infinite width. In a similar manner, one can define the bracket width of a Lie algebra. It is known that all finite-dimensional simple Lie algebras over an algebraically closed field have bracket width one. Our goal is to present first examples of simple Lie algebras of bracket width greater than one. This talk is based on a work in progress, joint with A. Regeta.