  It follows by Bixby's Lemma that if $e$ is an element of a $3$-connected
matroid $M$, then either ${\rm co}(M\delete e)$, the cosimplification of
$M\delete e$, or ${\rm si}(M/e)$, the simplification of $M/e$, is
$3$-connected. A natural question to ask is whether $M$ has an element $e$ such
that both ${\rm co}(M\delete e)$ and ${\rm si}(M/e)$ are $3$-connected. Calling
such an element "elastic", in this paper we show that if $|E(M)|\ge 4$, then
$M$ has at least four elastic elements provided $M$ has no $4$-element fans
and, up to duality, $M$ has no $3$-separating set $S$ that is the disjoint
union of a rank-$2$ subset and a corank-$2$ subset of $E(M)$ such that $M|S$ is
isomorphic to a member or a single-element deletion of a member of a certain
family of matroids.
