Consider the following fragments from Takesaki's second volume "Theory of operator algebras" (chapter VIII $\oint 3$, Modular Automorphism groups, p107-108:

Why are the second and third line in $(12)$ true? For example, I am not able to convince myself that we have the inclusion $$S(\mathfrak{D}^\sharp \cap \mathfrak{H}_1)\subseteq \mathfrak{H}_1.$$ Probably, some general fact about unbounded operators and their closures is used here but I am unable to identify what exactly I need.

Thanks in advance for any help/comments/insights!