  An element $e$ of a $3$-connected matroid $M$ is elastic if ${\rm si}(M/e)$,
the simplification of $M/e$, and ${\rm co}(M\backslash e)$, the
cosimplification of $M\backslash e$, are both $3$-connected. It was recently
shown that if $|E(M)|\geq 4$, then $M$ has at least four elastic elements
provided $M$ has no $4$-element fans and no member of a specific family of
$3$-separators. In this paper, we extend this wheels-and-whirls type result to
a splitter theorem, where the removal of elements is with respect to elasticity
and keeping a specified $3$-connected minor. We also prove that if $M$ has
exactly four elastic elements, then it has path-width three. Lastly, we resolve
a question of Whittle and Williams, and show that past analogous results, where
the removal of elements is relative to a fixed basis, are consequences of this
work.
