From the start, there's only one way you can go until you reach the water pool room. Take the path to the left, and when you reach the round lava pool, take a moment to look above it—there's a Fire Walk Scepter floating in the air over it. If you can use a fly spell, go ahead and grab it, then head around to the northwest section of the map, collecting the brass key on the way.
The black key is not strictly necessary, but makes the level more convenient to complete, so get it as well by jumping across the stone pillars, or flying there.
If you didn't get the Fire Walk Scepter, use a fire resistance potion, and go down into the big lava lake near where you got the keys. The pull chain on the north wall will make the pentagram accessible; grab it and head to the teleport on one of the stone islands in the lava.
Now you can head through the black key door to the south. The brass key door on the east side of the "four arrows" room leads to the exit.