The 2006 Cadillac DTS battery location is underneath the back seat’s bottom cushion. To get to it, you’ll need to pull up on the front of the cushion to release the front hooks.
Be sure to remove the vent tube connected to the battery carefully, and disconnect the black and then the red cables. Then, you’ll need a 13mm socket and an extension ratchet to unscrew the bolt holding in the battery.