The (weak) circuit elimination axiom states that if $C_1$ and $C_2$ are distinct circuits of a matroid and $e\in C_1\cap C_2$, then there exists a circuit $C_3$ such that $C_3\subseteq (C_1 \cup C_2) - e$.
I'm interested in the case where for two distinct circuits $C_1$ and $C_2$ with non-empty intersection there exists some $e\in C_1\cap C_2$ such that $C_3=(C_1 \cup C_2) - e$ is a circuit.
Under which conditions does $e\in C_1\cap C_2$ exist such that $C_3 = (C_1 \cup C_2) - e$ is a circuit?
Are there classes of matroids which satisfy this condition? For example, uniform matroids $U_{n+2}^{n}$ "trivially" satisfy it (and in fact they satisfy a stronger condition, namely elimination of any $e\in C_1\cap C_2$ will result in a circuit $(C_1 \cup C_2) - e$).