One of the motivation of the theory of Lie Algebras is that every associative algebra $A$ is a LA when the bracket is defined by $[a,b]=ab-ba$ : this is skew-symmetric and satisfies the Jacobi identity $[[a,b],c]+[[b,c],a]+[[c,a],b]=0$. Conversely, every abstract LA can be embedded into an associative algebra (its *envelopping algebra*). And for some good reason, one is really interested in sub-LAs rather than sub-algebras. A similar attitude, with different motivation lead to the notion of Jordan algebras.



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> If $A$ is an associative algebra, one may consider instead the *ternary* bracket $$[a,b,c]\_3=abc+bca+cab-acb-cba-bac.$$ 
>  Does $[.,.,.]\_3$ satisfy non-trivial identities, besides skew-symmetry? Is there any theory of abstract objects, vector spaces endowed which a ternary skew-symmetric product satisfying these identities?
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More generally, we may consider a $d$-bracket, which bears the name of *standard non-commutative polynomial* in $d$ non-commuting variables. For $d=2$, it is nothing but the standard bracket. When $d=2p$, the $d$-bracket does satisfy non-trivial identies, for instance
$$\sum\_{i\in\frak A\_7}[[a\_{i\_1},a\_{i\_2},a\_{i\_3},a\_{i\_4}],a\_{i\_5},a\_{i\_6},a\_{i\_7}]=0,\qquad\forall a\_1,\ldots,a\_7\in A.$$
I don't know if something non-trivial exists when $d$ is odd.



Ternary "Lie structure"
["Identities for the ternary commutator"](http://www.sciencedirect.com/science/article/pii/S0021869398974336) by Bremner classifies all such identities up to degree 7. A recent exposition can also be found in ["Ternutator Identities"](http://arxiv.org/abs/0908.1738) by C. Devchand, D. Fairlie, J. Nuyts, G. Weingart. Similar identities for n-ary commutators are proven in ["Multi-operator brackets acting thrice"](http://arxiv.org/abs/0905.2759) by T. Curtright, X. Jin, L. Mezincescu. I don't know if there is an accepted definition for what a Lie n-ary algebra should be for $n\geq 3$ (not to be confused with Lie n-algebras from nlab). For $n=3$ the most standard object one encounters are [Lie triple systems](http://en.wikipedia.org/wiki/Triple_system#Lie_triple_systems).


