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Lie Algebra**

A Lie algebra over a ring \(R\) is a left \(R\)-module \(\mathfrak{g}\) equipped with an \(R\)-bilinear map \([\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}\) which is alternating (\(\forall x\in\mathfrak{g}:[x,x]=0\)) and satisfies the Jacobi Identity (\(\forall x,y,z\in\mathfrak{g}:\big[x,[y,z]\big]+\big[z,[x,y]\big]+\big[y,[z,x]\big]=0\)).