Suppose $\mathbb{P}_0$, $\mathbb{P}_1$ and $\mathbb{P}_2$ are separative posets such that $\mathbb{P}_2$ projects into $\mathbb{P}_1$ and $\mathbb{P}_1$ projects into $\mathbb{P}_0$, i.e. there are projections of the form $\pi_{2,1}:\mathbb{P}_2\rightarrow\mathbb{P}_1$ and $\pi_{1,0}:\mathbb{P}_1\rightarrow\mathbb{P}_0$. For $i\in\{0,1,2\}$, let $\mathbb{B}_i$ be the boolean completion of $\mathbb{P}_i$. Using the composition $\pi_{2,0}:=\pi_{1,0}\circ\pi_{2,1}$, one can define a complete embedding $i$ from $\mathbb{B}_0$ to $\mathbb{B}_2$, by setting $$i(p_0)=\bigvee\{p_2\in\mathbb{P}_2\mid\pi_{2,0}(p_2)\leq p_0\},$$ for all $p_0\in\mathbb{P}_0$. I wonder whether this complete embedding admits some kind of factorization. More specifically, suppose $j:\mathbb{B}_0\rightarrow\mathbb{B}_1$ is an arbitrary complete embedding.
Question. Can we find a complete embedding $k:\mathbb{B}_1\rightarrow\mathbb{B}_2$ such that $k\circ j=i$?
I see the question is very general and the answer could be no, since there are no information on $j$. So I would like to see if the question has a positive answer under the following additional assumptions.
Let $\dot g_0$ be a $\mathbb{P}_1$-name such that $\Vdash_{\mathbb{P}_1}``\dot g_0$ is $\mathbb{P}_0$-generic$"$, let $G_2$ be $\mathbb{P}_2$-generic, and define $g_0=(\dot g_0)_{\pi_{2,1}``G_2}$ (i.e. $g_0$ is the interpretation of $\dot g_0$ by the $\mathbb{P}_1$-generic generated by $\pi_{2,1}``G_2$). Suppose there is a $\mathbb{P}_1$-generic $g_1\in V[G_2]$ such that $\pi_{1,0}``g_1\subseteq g_0$ (in other words, $g_1$ is $\mathbb{P}_1/g_0$-generic over $V[g_0]$). Define $$j(p_0)=\bigvee\{p_1\in\mathbb{P}_1\mid p_1\Vdash p_0\in\dot g_0\},$$ for all $p_0\in\mathbb{P}_0$. So now $j$ is no longer an arbitrary complete embedding, but a very specific one. The question is the same as above.
Question. Is there a complete embedding $k:\mathbb{B}_1\rightarrow\mathbb{B}_2$ such that $k\circ j=i$?
A candidate that I have in mind is $k(p_1)=\bigvee\{p_2\in\mathbb{P}_2\mid p_2\Vdash p_1\in \dot g_1\}$ for $p_1\in\mathbb{P}_1$, where $\dot g_1$ is a $\mathbb{P}_2$-name for $g_1$ (recall that $g_1\in V[G_2]$). However, I can't prove that the diagram commutes (i.e. I'm not sure $k\circ j=i$).
