Let $\left(g\_i\right)$ be a sequence of $N$ elements of a Lie algebra. Let $s$ be a cyclic permutation of $N$ elements of order $N$: $(1,2,...,N)\to(2,...,N,1)$. 


Let 
$c\_N=\sum\limits\_{k=1}^N[g\_{s^k(1)},[g\_{s^k(2)},[...,[g\_{s^k(N-1)},g\_{s^k(N)}]...]]]$.


We know that $c\_2=0$ by skew-symmetry and $c\_3=0$ by Jacobi identity. The question is: what restriction, if any, the condition $c\_N=0$ imposes on the Lie algebra?



Jacobi identity for circular permutations
The condition $c\_N=0$ does impose a restriction on a Lie algebra $L$ in general (with the exception of cases where, say, $L$ is nilpotent of class $c$, so that all terms of $c\_N$ are trivially zero for $N>c$). In the theory of identities one searches for a "basis" of identities. Yu P. Razmyslov found a finite basis for the identities of the simple Lie algebra $\mathfrak{sl}(2,K)$, see his article "Simple Lie algebras satisfying the standard Lie identity of degree $5$" and the references therein. For simple Lie algebras of rank at least $2$ (over an algebraically closed field of characteristic zero) an identity $c\_N=0$ should not hold for $n\ge 5$, I think.
For solvable and nilpotent Lie algebras these identities are more difficult to understand. Related to $c\_4=0$, but different, is the trilinear Malcev-identity
$$
[[x,y], [x,z]]=[[[x,y],z],x]+[[[y,z],x],x]+[[[z,x],x],y],
$$
which generalises the Jacobi identity. Another related topic consists of Filippov $n$-ary Lie algebras.


