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Abstract

The method of double extension, introduced by A. Medina and Ph. Revoy, is a procedure which decomposes a Lie algebra with an invariant symmetric form into elementary pieces. Such decompositions were developed for other algebras, for instance for Lie superalgebras and associative algebras, Filippov $n$-algebras and Jordan algebras.

The aim of this note is to find a unified approach to such decompositions using the derived bracket formalism. More precisely, we show that any commutative $n$-ary superalgebra with a skew-symmetric invariant form can be obtained inductively by taking orthogonal sums and generalized double extensions.