# Can algebras of elementary embeddings be sufficiently described by two element subalgebras?

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 $$\circ$$)?