How to include PyInstaller inside a PyInstaller bundle?

I want to freeze a Python application that has, as one of its features, the ability to produce freezed Python applications using PyInstaller. Here is a minimal application showing what I want to achieve:

import PyInstaller.__main__

with open('inception', 'w', encoding='utf-8') as f:
    f.write('import sys; print("Hello from the inside")\n')
PyInstaller.__main__.run(['--noconfirm', '--onedir', 'inception'])

Freezing this with PyInstaller

PS> pyinstaller --noconfirm --onedir example.py

should produce an executable exemple.exe that can be executed to produce inception.exe.

On the first try, I got the following error

PS> .\dist\example\example.exe
PyInstaller cannot check for assembly dependencies.
Please install pywin32-ctypes.

pip install pywin32-ctypes

This was fixed by installing pywin32 (pywin32-ctypes was already installed) and changing PyInstaller's compat.py file as explained here. Rebundling the application now results in the following error

PS> .\dist\example\example.exe
Traceback (most recent call last):
  File "example.py", line 2, in <module>
    import PyInstaller.__main__
  File "<frozen importlib._bootstrap>", line 983, in _find_and_load
  File "<frozen importlib._bootstrap>", line 967, in _find_and_load_unlocked
  File "<frozen importlib._bootstrap>", line 677, in _load_unlocked
  File "d:\code\stackoverflow\pyinstaller_inception\venv\lib\site-packages\PyInstaller\loader\pyimod03_importers.py", line 627, in exec_module
    exec(bytecode, module.__dict__)
  File "lib\site-packages\PyInstaller\__init__.py", line 66, in <module>
  File "lib\site-packages\pkg_resources\__init__.py", line 481, in get_distribution
  File "lib\site-packages\pkg_resources\__init__.py", line 357, in get_provider
  File "lib\site-packages\pkg_resources\__init__.py", line 900, in require
  File "lib\site-packages\pkg_resources\__init__.py", line 786, in resolve
pkg_resources.DistributionNotFound: The 'PyInstaller' distribution was not found and is required by the application
[14376] Failed to execute script example

So it seems like PyInstaller does not bundle itself inside the application. There is an issue on PyInstaller github page, but it does not really help. Is this even possible? If so, how?

This needs to run on Windows 10, with Python 3.7. I am using PyInstaller version 3.5.

from How to include PyInstaller inside a PyInstaller bundle?