The classical Yang-Baxter equation is \begin{align} [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0. \quad (1) \end{align} What are the differences between this equation in the case of Lie algebras ($r \in g \otimes g$, $g$ is a Lie algebra) and Lie superalgebras ($r \in g \otimes g$, $g$ is a Lie superalgebra)?

I am asking this question because I want to check an element $r$ in $g \otimes g$ ($g$ is a Lie superalgebra) satisfies (1) or not. In particular, I want to know the result of $[a \otimes b \otimes c, d \otimes e \otimes f]$, $a,b,c,d,e,f$ are in a Lie superalgebra $g$.

Thank you very much.


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