Classification of real Lie algebras of dimension one, two and three

Abstract

We present an exhaustive, self-contained, and coordinate-free structural classification of real Lie algebras of dimensions one, two, and three up to isomorphism. In dimensions one and two, the classification is governed entirely by the derived ideal and elementary derivation theory. In dimension three, we develop the classical exterior duality isomorphism $Λ^2\mathfrak{g} \cong \mathfrak{g}^* \otimes Λ^3\mathfrak{g}$ pioneered by Bryant and Milnor, which reduces the quadratic Jacobi identity on the $3 \times 3$ structure matrix $C = S + \hat{a}$ to the linear orthogonality relation $S \cdot a = 0$, where $a = \frac{1}{2}χ$ is the trace covector. We systematically bifurcate the three-dimensional category into unimodular ($a = 0$) and non-unimodular ($a \neq 0$) sectors, completely recovering the nine classical Bianchi isomorphism classes (Bianchi types I through IX) alongside their continuous moduli invariants. Furthermore, we establish the Levi--Malcev decomposition for low dimensions, compute the exact minimal representation dimension $μ(\mathfrak{g})$ via Ado's Theorem by constructing explicit matrix realizations in $\mathfrak{gl}(μ, \mathbb{R})$, and formulate a deterministic, coordinate-free decision algorithm for algebra identification.