Two groups of fire elementals stand on this bridge - on the far side, the swirling portal to Molten Core. This is where the quest can be completed - the core fragment is on the far side of the bridge, next to the portal. Note that you do not need to enter the core to finish this quest; in fact, if you enter the core by mistake, when you exit the core you will appear outside of the depths, next to the questgiver for the attunement quest. You will then need to run all the way back through the depths to rejoin your group.
The lava path shortcut
If you are aiming for The Emperor, Black Forge or the entrance to MC, you might consider taking the lava path. It begins below the Black Anvil, and ends at the Black Forge, bypassing much of the instance.
Jumping left after first entering the room from the direction of the bar into the lava its a short swim to the nearest flat piece of land that lines the walls. This path winds around until you come to a ledge far below the Chamber of Enchantment. Lower level players may need to deal with incidental elementals that flank this route on a small nearby island. The next step is a long stretch of lava patrolled by an elemental that you have to cross to reach an island in the center of the lava, which also has 4 elementals on it. Its best to pull the patrolling elemental and deal with it before crossing. It is possible to avoid all four elementals on the island, or to deal with them individually or in pairs. After that island, the last step is a short swim until you come up underneath the bridge, where you can climb up to the Black Forge. Beware of aggroing the 1st 3 elementals on the left.
The lava path allows for nice quick runs to 3 of the main objectives in BRD, but it can be dangerous for the inexperienced, or soloing characters. If you are aiming for the Emperor, you will come up before the Lyceum, needing the to continue, and you cannot move backwards into the room of 7. As described, it is possible currently for the 1st mob you fight in the instance to be the elemental patrolling the lava before the island.