  We study simply connected Lie groups $G$ for which the hull-kernel topology
of the primitive ideal space $\text{Prim}(G)$ of the group $C^*$-algebra
$C^*(G)$ is $T_1$, that is, the finite subsets of $\text{Prim}(G)$ are closed.
Thus, we prove that $C^*(G)$ is AF-embeddable. To this end, we show that if $G$
is solvable and its action on the centre of $[G, G]$ has at least one imaginary
weight, then $\text{Prim}(G)$ has no nonempty quasi-compact open subsets. We
prove in addition that connected locally compact groups with $T_1$ ideal spaces
are strongly quasi-diagonal.
