It follows by Bixbys Lemma that if $e$ is an element of a $3$-connected matroid $M$, then either $mathrm{co}(Mbackslash e)$, the cosimplification of $Mbackslash e$, or $mathrm{si}(M/e)$, the simplification of $M/e$, is $3$-connected. A natural question to ask is whether $M$ has an element $e$ such that both $mathrm{co}(Mbackslash e)$ and $mathrm{si}(M/e)$ 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.
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