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Jacobi Identity**

Given an abelian group \((A,+)\) and a binary product \([\cdot,\cdot]:A\times A\to A\) (typically \((A,+)\) has the additional structure of being a vector space and \([\cdot,\cdot]\) is also bilinear, but this is not required), one says that \([\cdot,\cdot]\) obeys the Jacobi identity if \(\forall x,y,z\in A:\big[x,[y,z]\big]+\big[y,[z,x]\big]+\big[z,[x,y]\big]=0\).