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The Jacobi identity is a stringent requirement on a possible set of structure constants to form a valid Lie algebra.

Extending or modifying this equation offers the possibility to discover new algebraic structures. One I know of leads to graded algebras, where one allows some changes in signs depending on the grading or commutator properties of the set of algebra generators:

$$(-1)^{ik}[x,[y,z]]+(-1)^{ij}[y,[z,x]]+(-1)^{ik}[x,[y,z]]+(-1)^{jk}[z,[x,y]]=0\,,$$ for generators $x,y$ and $z$ with respectively gradings $i,j$ and $k$.

(Note that I purposely rewrite history as I suppose that was not the initial motivation to consider graded algebras. But it can be defended from a didactical point of view ;-) .)

I am curious about other extensions or modifications of the Jacobi identity that lead to consistent algebraic structures?

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    $\begingroup$ The graded Jacobi identity is still the Jacobi identity, just with respect to a slightly exotic symmetric monoidal structure. You can write down the Jacobi identity in any symmetric monoidal linear category; one keyword to look up here is "operad." $\endgroup$ – Qiaochu Yuan Jul 30 '17 at 17:01
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One possible generalisation of the Jacobi identity is the Malcev identity $$ {\displaystyle (xy)(xz)=((xy)z)x+((yz)x)x+((zx)x)y,} $$ leading to Malcev-algebras. Another popular generalisation is the Hom-Jacobi identity, $$ [α(x),[y,z]]+[α(y),[z,x]]+[α(z),[x,y]]=0, $$

defining Hom-Lie algebras $L$, with $\alpha\in \operatorname{End}(L)$. There are many other modifications, i.e., with more terms and more variables, see here.

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