  We clarify the structure of nilpotent Lie groups which are multiplication
groups of $3$-dimensional simply connected topological loops and prove that
non-solvable Lie groups acting minimally on $3$-dimensional manifolds cannot be
the multiplication group of $3$-dimensional topological loops. Among the
nilpotent Lie groups for any filiform groups ${\mathcal F}_{n+2}$ and
${\mathcal F}_{m+2}$ with $n, m > 1$, the direct product ${\mathcal F}_{n+2}
\times \mathbb R$ and the direct product ${\mathcal F}_{n+2} \times _Z
{\mathcal F}_{m+2}$ with amalgamated center $Z$ occur as the multiplication
group of $3$-dimensional topological loops. To obtain this result we classify
all $3$-dimensional simply connected topological loops having a $4$-dimensional
nilpotent Lie group as the group topologically generated by the left
translations.
