I was too curious to wait for you to check if it works. So I tried and here is my result:
Yes, it works with the following recipe :-)
Create a live-only drive by cloning ubuntu-16.04.1-desktop-amd64.iso to a 32 GB USB 3 pendrive with mkusb
Add the boot option toram and boot
When running live, unmount all mounted partitions in the pendrive (in this case it was only one), run in a terminal window
sudo umount /media/{and press the TAB key for 'tab completion'}
# or
sudo umount /media/*/* # if more than one partition
Check that all partitions on the pendrive are unmounted with
df
Start the installer with the desktop icon
Follow the instructions like any other installation
Reboot
I did it in a laptop computer with 4 GiB RAM in BIOS mode without any internal drive.
If there is an internal drive, things may be more tricky in order to avoid writing bootloading things to the internal drive, but this is not due to the fact that it is installed to the same drive as the system was booted from.
-o-
Edit 1: I tested in UEFI mode. The final 'Reboot' got stuck at the plymouth 'five dots'.
and when rebooted, the installed system works. A better alternative is to
and it finishes and reboots beautifully.
-o-
Edit 2: You find help how to add the boot option toram at this link,
How to unmount a live DVD/USB?
Edit 3: Install Lubuntu Mantic to the same drive as you booted live from
Lubuntu uses the installer Calamares, and the version [of Lubuntu] is newer than what is described above, so there are a few more things to fix:
- boot with the boot options
toram
and nopersistent
- unmount the mounted partitions on the USB drive
- install
gparted
and create a fresh GUID partition table, GPT (no partition).
Now the installer is happy to install Lubuntu Mantic (23.10) into the USB drive, and the installed system works well.
Comment: I could have used an already available text mode tool to create a fresh GPT partition table, but gparted is convenient.
More details at Lubuntu Discourse