  Let $\widetilde{\mathcal M}=\langle \mathcal M, P\rangle$ be an expansion of
an o-minimal structure $\mathcal M$ by a dense set $P\subseteq M$, such that
three tameness conditions hold. We prove that the induced structure on $P$ by
$\mathcal M$ eliminates imaginaries. As a corollary, we obtain that every small
set $X$ definable in $\widetilde{\mathcal M}$ can be definably embedded into
some $P^l$, uniformly in parameters, settling a question from [10]. We verify
the tameness conditions in three examples: dense pairs of real closed fields,
expansions of $\mathcal M$ by a dense independent set, and expansions by a
dense divisible multiplicative group with the Mann property. Along the way, we
point out a gap in the proof of a relevant elimination of imaginaries result in
Wencel [17]. The above results are in contrast to recent literature, as it is
known in general that $\widetilde{\mathcal M}$ does not eliminate imaginaries,
and neither it nor the induced structure on $P$ admits definable Skolem
functions.
