I take it the LaCie is not self powered, right?
How is the drive connected to the Mac? Via a hub? If so are there other devices on the hub? Is the hub self powered?
Have you tried a different port and different cable?
Boot into Safe Mode according to How to use safe mode on your Mac and reboot normally and test.
NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following;
• Verifies your startup disk and attempts to repair directory issues, if needed
• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)
• Prevents Startup Items and Login Items from opening automatically
• Disables user-installed fonts
• Deletes font caches, kernel cache, and other system cache files
NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.