IMac won't shutdown USB disk

I didn't know where to ask this, but here seems like as good a place as any...
Yesterday I received a new iMac to replace my 2011 Mac Mini.  Attached to my Mini I had a Lacie Blade Runner 4TB USB drive.  When I power the Mini off, the Lacie powers off too.  And when I power on the Mini, the Lacie spins up and mounts.  Perfect.
I now have the LaCia attached to the iMac.  But when I shut the iMac down, the LaCie keeps spinning.  The only way to stop the disk spinning is to physically turn it off with the front mounted button.
Does anyone know what's going on and how I can get the LaCie to turn off when my iMac turns off?
Incidentally the LaCie *does* spin down when I put the Mac to sleep.  But not when I power it off.  Does that give anyone any clues?


I haven't excluded it from Spotlight search locations, so I would guess it is.  I do need to be able to search its contents.  Time Machine, no, I haven't enabled that (yet).
I did some further tests before bedtime and found that the disk will spin down whilst using the iMac, in line with the "Put hard disks to sleep when possible" option, which is checked.
If I eject the drive, it will spin down as well.  But then if I shutdown the iMac (with the disk still ejected) it spins up the disk as part of the shutdown process and it stays spinning with the iMac switched off.
Very puzzling and a complete pain in the ***!