Standard systems for semifinite O$^*$-algebras

We shall continue the study of standard systems which make it possible to develop the Tomita-Takesaki theory in O$^*$-algebras. The main purpose of this paper is to give the necessary and sufficient conditions for which a standard system $({\cal M}, \lambda , \lambda ')$ of an O$^*$-algebra ${\cal M}$, a generalized vector $\lambda$ and the commutant $\lambda '$ is unitarily equivalent to a standard system $\bigl ( {\cal N}, K' \mu , (K' \mu )'\bigr )$ constructed by a standard tracial generalized vector $\mu$ for an O$^*$-algebra ${\cal N}$ and a non-singular positive self-adjoint operator $K'$ affiliated with the commutant ${\cal N}'_{ \mathrm {w}}$ of ${\cal N}$.