Pull the underwater lever, turn around and swim to the ledge at water level just below the tall pillar near the entrance. Climb out of the water onto the ledge so you're already facing the pillar. Then when you climb up, you're ready to take the next jump without having to set up first.
Take a running jump over the spikes to the nearest flat spot. Press the right direction key to pivot in the air so Lara lands on the flat spot facing the far wall. If you make it this far without stumbling, save your game if possible. Then, if you mess up, you can just reload and skip the business with the lever and the swimming.
Step back until her heels touch the raised edge of the tile behind her. Then take a running jump straight across the pyramid. Lara should land on the slippery slope and slide down to the flat spot above and beyond the second spike pit.
Again, step back so Lara has enough distance for the take-off and do another running jump to the ledge near the red door. You should have plenty of time to go through before it closes. Just be careful not to run on into the lava in the room ahead.
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