  In this paper we deal with the class C of decomposable solvable Lie groups
having dimension at most six. We determine those Lie groups in C and their
subgroups which are the multiplication group Mult(L) and the inner mapping
group Inn(L) for three-dimensional connected simply connected topological loops
L. These loops L have one- or two-dimensional centre and their group Mult(L)
has two- or three-dimensional commutator subgroup. Together with this result we
obtain that every at most 3-dimensional connected topological proper loop
having a solvable Lie group of dimension at most six as its multiplication
group is centrally nilpotent of class two.
