Akivis algebra

In mathematics, and in particular the study of algebra, an Akivis algebra is a nonassociative algebra equipped with a binary operator, the commutator formula_1 and a ternary operator, the associator formula_2 that satisfy a particular relationship known as the Akivis identity. They are named in honour of Russian mathematician Maks A. Akivis.
Formally, if formula_3 is a vector space over a field formula_4 of characteristic zero, we say formula_3 is an Akivis algebra if the operation formula_6 is bilinear and anticommutative; and the trilinear operator formula_7 satisfies the "Akivis identity":
An Akivis algebra with formula_9 is a Lie algebra, for the Akivis identity reduces to the Jacobi identity. Note that the terms on the right hand side have positive sign for even permutations and negative sign for odd permutations of formula_10.
Any algebra (even if nonassociative) is an Akivis algebra if we define formula_11 and formula_12. It is known that all Akivis algebras may be represented as a subalgebra of a (possibly nonassociative) algebra in this way (for associative algebras, the associator is identically zero, and the Akivis identity reduces to the Jacobi identity).