Detachable pairs in $3$-connected matroids and simple $3$-connected graphs

Let $M$ be a $3$-connected matroid. A pair $\{e,f\}$ in $M$ is detachable if $M\mathrm{\backslash }e\mathrm{\backslash }f$ or $M\mathrm{/}e\mathrm{/}f$ is $3$-connected. Williams (2015) proved that if $M$ has at least 13 elements, then at least one of the following holds: $M$ has a detachable pair, $M$ has a $3$-element circuit or cocircuit, or $M$ is a spike. We address the case where $M$ has a $3$-element circuit or cocircuit, to obtain a characterisation of when a matroid with at least 13 elements has a detachable pair. As a consequence, we characterise when a simple $3$-connected graph $G$ with $\mathrm{\mid }E(G)\mathrm{\mid }\ge 13$ has a pair of edges $\{e,f\}$ such that $G\mathrm{/}e\mathrm{/}f$ or $G\mathrm{\backslash }e\mathrm{\backslash }f$ is simple and $3$-connected.