Abstract: Among all Frobenius Lie groups having a complement isomorphic either to $\CC^\times$ or to $\HH^\times$ and a kernel which is a vector group those are determined that admit a planar partition into closed subgroups. Moreover, it is shown that for each of these groups the exponential function induces a bijection between the set of planar partitions of the group and the set of planar partitions of the associated Lie algebra.
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