Upgrading from 64-bit to 32-bit Windows 10.
 

The OS on this machine is Windows 10 home 64-bit, installed by the machine
manufacturer. I want to install Windows 10 home 32-bit in its place, which
means I need the corresponding ISO image, which won't be of any use if I
don't give the installer the product key (PK).

The disk space required by the 32-bit version is 16GB and I'm in need of
space to make room for Linux while still keeping Windows to keep in touch
with tech forums while I get a running Linux GUI [1]. I would dedicate the
thread to the product key problem. By searching in Google I learn a new
installatioon is mandatory. Microsoft says something like: if you bought
a machine running Windows 10 the PK is on the box. But I bought the machine
second hand and don't have the box.

One question therefor is: How do I get the PK? I think either the OS gives me
the PK or Microsoft does. Another one: assuming I've got it, how do I get
an ISO image that will ask for precisely that PK. As I have put it the
situation is helpless, but perhaps I do not have the correct perspective.

[1] If anything goes wrong during Linux installation I risk loosing connection
with the forums. I intend to install Arch Linux, and from the Arch install
proper to having a running GUI (and therefor forums) is a long way, I under
stand. Please don't make this the thread subject.

PS: Activation helps verify that your copy of Windows is genuine and hasn’t been used on more devices than the Microsoft Software License Terms allow. Depending on how you got your copy of Windows 10, activation will use either a digital license or a 25-character product key. A digital license (called a digital entitlement in Windows 10, Version 1511) is a method of activation in Windows 10 that doesn't require you to enter a product key.

During installation, you’ll be prompted to enter a product key if no digital license is found on your computer. A valid product key or digital license is required to activate Windows 10; without one, activation can’t continue.

To find the activation method that works with your copy of Windows 10, see the table below.
https://support.microsoft.com/en-us/help/12440

Depends on who made the system. I think lenovo has it in bios. You use a linux command to recover it.

Magicjellybean may still work.

Magicjellybean gives
As a product key should have five fields, I assume it is the CD key above. Am I right? Thanks for your post.

The CD key looks like a Windows key.

Instead of worrying about losing your MS windows 10 why not just buy another hard disk. They're not very expensive. You can then install linux on the new disk and keep your windows 10 64bit.