Three-dimensional topological loops with solvable multiplication groups

We prove that each $3$-dimensional connected topological loop $L$ having a solvable Lie group of dimension $\le 5$ as the multiplication group of $L$ is centrally nilpotent of class $2$. Moreover, we classify the solvable non-nilpotent Lie groups $G$ which are multiplication groups for $3$-dimensional simply connected topological loops $L$ and $\hbox{dim} \ G \le 5$. These groups are direct products of proper connected Lie groups and have dimension $5$. We find also the inner mapping groups of $L$.