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.

If $A$ is an associative algebra, one may consider instead the

ternarybracket $$[a,b,c]_3=abc+bca+cab-acb-cba-bac.$$ 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?

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.

thirdsone is the quotient of the free algebra on three generators $a$, $b$, $c$ modulo what you write as $[abc]_3$; the algebra is encodes then properties of triads of elements which "ternary-Lie-commute". The algebra is $3$-Koszul in the sense of Berger, so quite special: maybe you can extract information from that fact) – Mariano Suárez-Alvarez♦ May 10 '12 at 18:21