Is $vN(M_1,M_2) \cap M_3= vN(M_1,M3) \cap vN(M_2,M3)$?

Question

Let $M_1,M_2,M_3$ be von Neumann algebras (i.e. weakly closed subalgebras of $B(H)$ where $H$ is a Hilbert space).

Let $vN(M_1,M_2)$ denote the von Neumann algebra generated by $M_1$ and $M_2$ inside $B(H)$.

Is it true that $vN(M_1,M_2) \cap M_3= vN(M_1,M3) \cap vN(M_2,M3)$?

Answer 1

Let $M_1=M_2=M_2(\mathbb C)$, and $M_3=\mathbb C\,I_2$. Then $$ vN(M_1,M_2)\cap M_3=\mathbb C\,I_2, $$ $$ vN(M_1,M_3)\cap vN(M_2,M_3)=M_1\cap M_2=M_2(\mathbb C). $$