  We give a number of equivalent conditions (including weak centrality) for a
general $C^*$-algebra to have the centre-quotient property. We show that every
$C^*$-algebra $A$ has a largest weakly central ideal $J_{wc}(A)$. For an ideal
$I$ of a unital $C^*$-algebra $A$, we find a necessary and sufficient condition
for a central element of $A/I$ to lift to a central element of $A$. This leads
to a characterisation of the set $V_A$ of elements of an arbitrary
$C^*$-algebra $A$ which prevent $A$ from having the centre-quotient property.
The complement $\mathrm{CQ}(A):= A \setminus V_A$ always contains
$Z(A)+J_{wc}(A)$ (where $Z(A)$ is the centre of $A$), with equality if and only
if $A/J_{wc}(A)$ is abelian. Otherwise, $\mathrm{CQ}(A)$ fails spectacularly to
be a $C^*$-subalgebra of $A$.
