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About a year and a half ago, I was at a talk by Martin Hyland where he suggested that the Jacobi identity is to the associative law as the anticommutative law is to the commutative law. I think this was in the context of some kind of duality for operads, but I didn't understand at the time.

More recently, I've come to understand that the associative law and the Jacobi identity are essentially the same in the following sense: they both make self-action representations possible. Indeed, the associative law says $$(x \cdot y) \cdot {-} = (x \cdot {-}) \circ (y \cdot {-})$$ and the Jacobi identity says $$[[x, y], {-}] = [[x, {-}], [y, {-}]]$$

Question. Is there a way to make this precise in the language of universal algebra or (enriched) category theory, and are there other examples?

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    Quibble: the fact that you use $[ \cdot, \cdot ]$ for both the Lie bracket on $\mathfrak{g}$ and the commutator on $\text{End}(\mathfrak{g})$ is mildly confusing, since it looks like you could be taking the Lie bracket of $[x, z]$ and $[y, z]$ pointwise.2012-06-25
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    I am amused to discover that you had asked the [same question](http://mathoverflow.net/questions/21152) some years earlier.2014-02-09

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