lsof reports no files open on external volume but it still cannot be safely unmounted

I've searched around here on SX and other places but have not found an approach that gives me any traction on my issue. I've a volume (used by Time Machine as it happens) on an external Firewire drive. When the drive is connected, it has 3 volumes. Two out of three eject with no problems but the last one (LaCie) can never be unmounted unless by force (e.g. $ umount -f LaCie). I can see the drive being accessed almost continuously but I have no idea which process is doing this; and lsof is of limited help. Mostly it returns no results as below but occasionally it reports that Finder has a directory open on the volume. But I don't see why that would be a problem as Finder should close any open files or directories when a volume is unmounted. This problem only arose after I upgraded to Mountain Lion. I use a cloning tool (SuperDuper!) but have made sure that is not running; and lsof appears to confirm this. How else can I find out what is locking up the volume? I'm not keen on forcibly ejecting the volume every time I want to disconnect the drive as I surely risk data loss on my main backup disk.

~ $ sudo diskutil unmount /Volumes/LaCie/ 
Volume LaCie on disk1s1 failed to unmount 
~ $ lsof | grep LaCie
~ $ lsof | grep /Volumes/LaCie
~ $

If you run lsof without sudo, it can only see what files your processes have open, not those open by system or other users. Use sudo lsof | grep /Volumes/LaCie to get a better idea what's going on.