Decisive subgroups of analytic groups

It is known that every analytic group $(L,\tau )$ contains a closed abelian subgroup $H$ which is "decisive" in the sense that the Hausdorff topologies for $L$ which are weaker than $\tau$ are completely determined by their restrictions to $H$. We show here that $H$ must ordinarily contain the entire center of $L$ but that the rest of $H$ can in general be reduced. The proof involves constructing "unusual" topologies for abelian Lie groups.