Suppose $M$ is $II\_{1}$ factor but need not be in standard form. Under what condition (on $M$ or Hilbert space) is the commutant $M'$ of $M$ again $II\_{1}$ factor on the Hilbert space acted by $M$?



On commutant of $II_{1}$ factors
Look at V. Jones 2015 von Neumann notes
<https://math.vanderbilt.edu/jonesvf/VONNEUMANNALGEBRAS2015/VonNeumann2015.pdf>


Theorem 10.2.1(1). 


You shall learn the coupling constant first.


