Double extension for commutative $n$-ary superalgebras with a skew-symmetric invariant form
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.