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A well-known result on Lie Theory states that every finite-dimensional complex solvable Lie algebra can be represented as a matrix Lie algebra, with upper-triangular square matrices as elements. However, this result does not specify which is the minimal order of the matrices involved in such representations. Hence, the main goal of this paper is to revisit and implement a method to compute both that minimal order and a matrix representative for a given solvable Lie algebra. As application of this procedure, we compute representatives for each solvable Lie algebra with dimension less than $6$., 19 pages, 6 tables

According to Ado and Cartan Theorems, every Lie algebra of finite dimension can be represented as a Lie subalgebra of the Lie algebra associated with the general linear group of matrices. We show in this paper a method to obtain the simply connected Lie group associated with a nilpotent Lie algebra, by using unipotent matrices. Two cases are distinguished, according to the nilpotent Lie algebra is or not filiform.

This work shows how to associate the Lie algebra h n , of upper triangular matrices, with a specific combinatorial structure of dimension 2 , for n ∈ N . The properties of this structure are analyzed and characterized. Additionally, the results obtained here are applied to obtain faithful representations of solvable Lie algebras.