Classification of real Lie algebras of dimension four

Abstract

We present an exhaustive, coordinate-free, and self-contained classification of four-dimensional real Lie algebras up to isomorphism. In dimension three, classification schemes frequently exploit Bryant's classical exterior duality isomorphism $Λ^2\mathfrak{g} \cong \mathfrak{g}^* \otimes Λ^3\mathfrak{g}$, which reduces the Jacobi identity to an elegant linear algebraic condition on a single $3\times 3$ matrix. We explicitly demonstrate that this accidental isomorphism breaks down in dimension four due to the dimensional mismatch $\dim Λ^2\mathfrak{g} = 6 \neq 4 = \dim(\mathfrak{g}^* \otimes Λ^4\mathfrak{g})$, transforming the Jacobi identity into a non-linear system of 16 quadratic constraints on 24 structure constants. To overcome this geometric obstruction without relying on brute-force polynomial elimination, we employ an intrinsic Lie-theoretic reduction: the Levi--Malcev decomposition isolates the non-solvable sector into direct products with three-dimensional simple ideals, while the solvable sector is classified by anchoring the structure to its nilradical $\mathfrak{n}$, which is an intrinsic characteristic ideal. For codimension-one nilradicals ($\dim\mathfrak{n} = 3$), the problem is reduced to classifying outer derivations of $\mathbb{R}^3$ and the Heisenberg algebra $\mathfrak{h}_3$ under the action of the automorphism group modulo global rescaling and central translations. For two-dimensional nilradicals, the structure is resolved via maximal abelian sub-algebras of $\mathfrak{gl}(2,\mathbb{R})$ lacking nilpotent elements. Complete proofs are supplied for all reductions, precise moduli parameter ranges are established to eliminate redundancies, and an explicit isomorphism dictionary is provided bridging our structural models with classical tabulations in mathematical physics.