Cannot Unmount Mac OS X Base System Disk
The Mac OS X base system image is not the cause of your problem. See What is the 'Mac OS X Base System' disk image on my 2011 MacBook AIr? for an explanation.

My best guess, based on the screenshot, as to why you can't repartition the Hitachi disk is that it's the current startup device. You need to restart your computer from an an external device—preferably a device that has the Lion Installer on it-if you want to reformat the drive. Otherwise, assuming you have a valid recovery partition, you can erase the Mac partition itself and reinstall Lion without an external device.

Perhaps Apple's Mac OS X 10.7 Help: Recover your entire system document will help you.