The Lie Bianchi classification of 3 dimensional Lie algebras continues to be of current interest – as in here https://arxiv.org/pdf/1403.2278.pdfbecause of its connections with classification problems in geometry and differential equations.
I will outline a proof that works for higher dimensional algebras. It is based on Lie’s theorem for solvable Lie algebras and the main technical difficulty is to work with the Jordan form of more than one transformation. I will also explain how one can determine canonical representatives as vector fields in the plane and in R^3. This is the main technical tool for obtaining canonical forms of invariant equations. If the invariant equations are linear for a certain Lie algebra of vector fields, then any equation with that symmetry algebra is linearisable.
