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

[![enter image description here][1]][1]

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/b8ioR.png
  [2]: https://i.sstatic.net/TMo9w.png

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 your help!