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Given an associative algebra \(\mathfrak{A},\diamond\) over a ring \(R\), the commutator Lie algebra of \(\mathfrak{A}\) is the Lie algebra \(\mathfrak{A},[\cdot,\cdot]_\diamond\) with bracket \([x,y]_\diamond:=x\diamond y-y\diamond x\). The mapping of an associative algebra to its commutator algebra is functorial and is right adjoint to the functor associating an \(R\)-Lie algebra to its universal enveloping algebra.