Let $N\subset M$ be any irreducible (i.e.$ N'\cap M=\Bbb C 1$) inclusions of factors with separable predual and with expectation. Popa proved that if $N$ is semifinite, then there exists an AD subfactor with expectation $P\subset N$ such that $P'\cap M=\Bbb C 1$.
Can we remove the condition that $N$ is reducible in $M$?
More precisely, if $N\subset M$ is any inclusion of factors with separable predual and with expectation. Suppose that $N$ is semifinite, does there exists an AFD subfactor with expectation $P\subset N$ such that $P'\cap M=\Bbb C 1$?
Since $P\subset N$, you have that $N'\subset P'$. So if $P'\cap M=\mathbb C$, then $$ \mathbb C\subset N'\cap M\subset P'\cap M=\mathbb C. $$ That is, $N'\cap M=\mathbb C$. In other words, if $N$ is not irreducible then what you want is impossible.