Matrix Representation for Decomposable Solvable Six-Dimensional Lie Algebra

This paper extends the classification of three-dimensional connected topological proper loops , for which the multiplication group is a six-dimensional decomposable solvable Lie group. Building upon the significant results presented in [1], which established class two central nilpotency for these loops, we derive explicit matrix representations for the associated Lie algebras. These representations are critical for completing the classification, as they facilitate verification of structural compatibility with the conditions dictated by the Lie bracket. We identify distinct families of matrix Lie groups, including groups characterized by one-dimensional centers and groups with two-dimensional centers. Additionally, we clarify the correspondence between these groups and their derived Lie algebras. Our findings address an existing gap in the literature by providing a systematic framework for examining the interrelationships between topological loops and their corresponding Lie groups.