Troubleshooting: 90 seconds to eject external Time Machine drive?
Some process (or processes) seem to be accessing the drive and preventing it from ejecting. You can list those processes with this command:
sudo lsof +d /Volumes/&lt;Time Machine hard drive name&gt;

which displays all running processes that are accessing the folder /Volumes/&lt;Time Machine hard drive name&gt;
From the information in your post, the culprit is mds:
sudo lsof +d /Volumes/LaCie.Backup
COMMAND PID USER   FD   TYPE DEVICE SIZE/OFF NODE NAME
mds      90 root   21r   DIR   1,16      510    2 /Volumes/LaCie.Backup
mds      90 root   25r   DIR   1,16      510    2 /Volumes/LaCie.Backup
mds      90 root   27r   DIR   1,16      510    2 /Volumes/LaCie.Backup
mds      90 root   31r   DIR   1,16      510    2 /Volumes/LaCie.Backup
mds      90 root   34r   DIR   1,16      510    2 /Volumes/LaCie.Backup

which is the metadata server used by Spotlight.
To prevent this from happening, add the Time Machine drive to the Spotlight exceptions:

Open System Preferences &gt; Spotlight.
Select the Privacy tab.
Press the + button, select the Time Machine drive and select &quot;Choose&quot;.

You may need to reboot your Mac for changes to take effect.