Isomorphism theorem on low dimensional Lie algebras (Q706060)

For any Lie algebra \(L\) denote its universal associative envelope by \(U(L)\). The authors prove the following theorem: Let \(L\) be a finite-dimensional Lie algebra over a field \(k\) of characteristic not 2, and let \(M\) be a 3-dimensional Lie algebra over \(k\). Then \(L\) is isomorphic to \(M\) if and only if \(U(L)\) is isomorphic to \(U(M)\). -- For 1- or 2-dimensional Lie algebras this follows from the classification [see \textit{N. Jacobson}, Lie algebras. Intersci. Tracts in Pure and Appl. Math. 10 (New York: Intersci. Publishers) (1962; Zbl 0121.27504)], for dimension 3 it was proved by [\textit{P. Malcolmson}, J. Algebra 146, No. 1, 210--218 (1992; Zbl 0752.17009)]; the authors give a proof by describing the simplicity of \(M\) in terms of \(U(M)\) and then proving the result when \(M\) is not simple.