The multiplication groups of 2-dimensional topological loops

If the multiplication group Mult(L) of a connected simply connected 2-dimensional topological loop L is a Lie group, then Mult(L) is an elementary filiform Lie group F of dimension n + 2 for some n >= 2, and any such group is the multiplication group of a connected simply connected 2-dimensional topological loop L. Moreover, if the group topologically generated by the left translations of L has dimension 3, then L is uniquely determined by a real polynomial of degree n.