With or without the PYTHON variables defined, my ArcMap/ArcCatalogue opens. But I tried it anyway. The thing is, you have to point to the 32-bit version. ( C:\Python27\ArcGIS10.4 ) even if you have a 64-bit machine. Pointing to the 64-bit version, python command line in ArcMap/ArcCatalogue will not work. Being still a native 32-bit app, I think ArcMap/ArcCatalogue still uses Python 32-bit version. Wonder what is the use of the Python 64-bit version. ( C:\Python27\ArcGISx6410.4)
So far, I have done;
1. Resetting ArcGIS Profile ( registry , file system )
2. Define PYTHON variables ( PYTHONHOME, PYTHONPATH )
3. Reset IE 11 default settings ( some setting controlled by sys adm )
4. As defined in KB 41057
Just for added info.
1. Recently installed some Windows Update ( no control on this... IT Policy)
2. ArcGIS Pro 1.0 installed
3. ArcGIS Earth installed
No success 😞