The multiplication groups of 2-dimensional topological loops

We prove that if the multiplication group $Mult(L)$ of a connected $2$-dimensional topological loop is a Lie group, then $Mult(L)$ is an elementary filiform nilpotent Lie group of dimension at least $4$. Moreover, we describe loops having elementary filiform Lie groups $\mathbb F$ as the group topologically generated by their left translations and obtain a complete classification for these loops $L$ if $\hbox{dim} \ \mathbb F=3$. In this case necessary and sufficient conditions for $L$ are given that $Mult(L)$ is an elementary filiform Lie group for a given allowed dimension.