Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$. Then $\mathcal{E}_{\lambda}$ can be endowed with a self-distributive operation $*$ where $j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$.
Suppose that $j_{1},...,j_{n}\in\mathcal{E}_{\lambda}$. Then do there exist some $j,k,l\in\mathcal{E}_{\lambda}$ where $j*j_{1},...,j*j_{n}\in\langle k,l\rangle$ (I am slightly more interested in when the closure is the closure only under $*$ instead of the closure under both $*$ and $\circ$)?
