The three-dimensional Lie algebras of locally transitive transformations of space (Q1387283)

A complete classification of real 3-dimensional abstract Lie algebras was obtained by Bianchi in 1918. Each of these Lie algebras can be described by the multiplication table of a standard basis \(\{X_1, X_2, X_3\}\) [cf. \textit{A. Z. Petrov}, New methods in general relativity theory (Russian). Moscow: Nauka (1966; Zbl 0146.23901)]. In the article under review, it is shown that, as the basis of a Lie algebra of a local Lie group of locally transitive transformations of \(\mathbb{R}^3\), \(\{X_i,\;i=1,2,3\}\) can be expressed explicitly in a definite form of differential operators with respect to an approximate coordinate system.