I have often read the term free element of a given matroid $M$. However, I could not find a proper defintion of what a free element actually is. I know what the free matroid is but free elements seem to be something else. I first thought that free elements are coloops. However, I have read sentences like "then $e$ must be a coloop or a free element of $M$" which suggests that coloops and free elements are not necessarily the same.
For example, free elements are mentioned in this paper on pages 4 and 11: https://www.math.lsu.edu/~oxley/LaminarMatroids_revised.pdf
I would appreciate if someone could give me a definition of a free element or/and provide an example of such an element in a certain matroid.
An element $e$ in $M$ is free if it is not a coloop(not all basis contain $e$) and if $e\in C$ circuit, then there is a basis $B$ such that $C=C(B,e)$, that is a spanning circuit.
For example, consider $M(C_n)=M(\{0,\cdots ,n-1\},\{\{i,(i+1)\pmod n\}\})$ the cycle matroid of a circuit with $n$ elements. There is just one circuit and it is spanning. For any edge $e$, there is a basis that does not contain $e$ (mainly $\{0,\cdots ,n-1\}\setminus e$), so every element is free there.
Check the definition here (Unbreakable Matroids).