Abstract. If the multiplication group MultðLÞ of a connected simply connected 2-dimensional topological loop L is a Lie group, then MultðLÞ is an elementary filiform Lie group F of dimension n þ 2 for some n d 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.

Our aim in this paper is to classify the 3-dimensional connected differentiable global Bol loops, which have a non-solvable group as the group topologically generated by their left translations and to describe their relations to metric space geometries.

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