We prove that the solvability of the multiplication group Mult(L) of a connected simply connected topological loop L of dimension three forces that L is classically solvable. Moreover, L is congruence solvable if and only if either L has a non-discrete centre or L is an abelian extension of a normal subgroup ℝ by the 2-dimensional nonabelian Lie group or by an elementary filiform loop. We determine the structure of indecomposable solvable Lie groups which are multiplication groups of three-dimensional topological loops. We find that among the six-dimensional indecomposable solvable Lie groups having a four-dimensional nilradical there are two one-parameter families and a single Lie group which consist of the multiplication groups of the loops L. We prove that the corresponding loops are centrally nilpotent of class 2.

我们证明了三维力的连接的简单连接拓扑回路L的乘法群Mult（L）的可解性，L是经典可解的。此外，当且仅当L具有非离散中心或L时，L是同余可解的是正常子组ℝ的二维扩展非阿贝尔李群或基本丝状环的阿贝尔扩展。我们确定不可分解的可解李群的结构，它们是三维拓扑回路的乘法群。我们发现，在具有四维零基的六维不可分解的可解李群中，有两个一参数族和一个由环L的乘法群组成的李群。我们证明了相应的循环在类2的中央幂等。