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Let $A$ a algebra over a field $K$. We define the associator of elements $a,b,c \in A$ as $$(a,b,c) = (ab)c - a(bc).$$ We define the commutator too by $$[a,b] = ab - ba.$$

I'm looking for some relationship between them, more precisely to try to solve a Jordan algebra problem that I already know has a solution for Lie algebras.

I researched some time and I didn't find a reference that deals with this subject and more, I don't even know if there is such a relationship.

I decided to ask here, because sometimes someone has already worked or seen something in that sense.

Thank you very much in advance.

Croos
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  • When I searched "commutator associator relationship" i got this hit: [Associator-Commutator Identity](https://math.stackexchange.com/q/1180393/29335) if that is helpful in your context. – rschwieb Nov 05 '20 at 19:15

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In any algebra over $K$ we have the following relationship:

\begin{align*} [[a,b],c]+[[b,c],a]+[[c,a],b] & = (a,b,c)+(b,c,a)+(c,a,b)-(b,a,c)-(a,c,b)-(c,b,a), \end{align*}

This is of course in the context of the Jacobi identity for Lie algebras. But there are several more such identities, also in the context of Jordan algebras (with $[[x,y]]=xy+yx$).

Dietrich Burde
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