  By [6], a minimal group $G$ is called $z$-minimal if $G/Z(G)$ is minimal. In
this paper, we present the $z$-Minimality Criterion for dense subgroups with
some applications to topological matrix groups. For a locally compact group
$G$, let $\operatorname{Inn}(G)$ be the group of all inner automorphisms of
$G,$ endowed with the Birkhoff topology. Using a theorem by Goto [14], we
obtain our main result which asserts that if $G$ is a connected Lie group and
$H\in\{G/Z(G), \operatorname{Inn}(G)\},$ then $H$ is minimal if and only if it
is centre-free and topologically isomorphic to $\operatorname{Inn}(G/Z(G)).$ In
particular, if $G$ is a connected Lie group with discrete centre, then
$\operatorname{Inn}(G)$ is minimal. We prove that a connected locally compact
nilpotent group is $z$-minimal if and only if it is compact abelian. In
contrast, we show that there exists a connected metabelian $z$-minimal Lie
group that is neither compact nor abelian.
