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?