I got stuck when reading Apendix II of G-Induced Flows by L. Auslander and L. Green and hopefully someone could help me out. The article can be found at https://www.jstor.org/stable/2373046 and let me describe the setting of my problem.
Let $S_2$ a three dimensional sovable Lie group whose Lie algebra has a basis $\{X,Y,Z\}$ with relations $[X,Z] = \sigma Y + Z, [X,Y] = Y - \sigma Z, [Y,Z]=0$, where $\sigma$ is a nonzero real number. The author then represents $S_2$ by a pair $(t,w)$, $t$ real, $w$ complex with the product relation $$(t,w)(t',w') = (t+t',w + e^{(1-i\sigma)t}w' )$$
I wonder how this representation of $S_2$ is obtained. Are three dimensional solvable Lie group actually classifiable so that the author just identify $S_2$ by its Lie algebra and that the representation above is another well-known model for this Lie group? Or is this representation derived purely from the Lie algebra structure of $S_2$?
Any comment is highly appreciated!