Republic Fleet
The crystal is already in the Fleet, it is just a matter of traveling to the Museum. The reason you need the Cormium Crystal Shard is because this will open the door to gain access for the +10 All Datacron.
When you use your map, click your smaller map to extend it out and you should see the 3 Main ships that we are able to travel to. If you have Quick Travel available and have discovered Gav Daragon’s Quick Travel, then you can use your QT but remember you still have to use the elevator to transport to Gav Daragon Bridge Deck.
If you don’t have QT available, use your speeder: From the Fleet, open your map and head East. The elevator should be called Mission Departures.
From the elevator click Interfleet Transport.
Use the Republic Fleet Transport (the Transport Service) which this Level 50 Vendor droid C2-look-alike will shuttle you to Gav Daragon.
Once you have arrived, there is another elevator so click its button to transport yourself to Gav Daragon Bridge Deck.
Looking on your Map again, this is your third and last elevator to use. It is south from what the map shows, which this will lead you to the Museum.
When inside the Museum…
Museum
Your map is seen as The Gav Daragon: Republic Museum.
You should see a small machine, that-sort-of-looks-like-one-but-not-really Walker. It is kind of tricky to jump on top of the ledge but there are a few sweet spots to easily reach over the ledge. Here is one of my spots that I suggest to try, but it is not the only way to do a successful jump. You can jump with or without a speeder, since it is more focus upon how far you can jump across while you must continue moving forward as you jumped up.
Go inside the machine and there should be a little panel, called a Weapons Console.
When you click it, you might hear or barely hear an explosion. Thus this opens the door which leads to the Cormium Crystal.
Go around the small machine, heading West by reflecting on the Map and you will see an open door. Run to it since there is a timer to remain open.
Once inside the tunnel, take a left.
You will take another left through the small tunnel.
Your last left turn should appear the large blue crystal which you have seen earlier.
Go to the right side of the Crystal, there should be a clickable part of it.
Once you click it, a shard should be in your Mission Items (tab) from your Inventory. It is called a Cormium Crystal Shard. AKA Blue Crystal.
To Exit out of the area, retrace your path which you have recently done and click the switch to open the door. You also can open the door for others too!
* HOW TO BYPASS THROUGH THE DOOR WITHOUT THE RED CRYSTAL
To start and enter through the door for the +10 All Datacron. There is a shielded door which is right across the statue that is used by the Corellia Crystal (Red Crystal) to open this door but if someone is already behind it than he or she is able to re-open the door without the use of the Corellia Crystal.
KEEP IN MIND YOU ARE RELYING ON A PLAYER TO OPEN THE DOOR FOR YOU! SO IF YOU FALL OFF FROM ANY LEDGE THAN YOU WILL NEED SOMEONE TO CONTINUE OPENING THE DOOR FOR EVERY FAILED ATTEMPT YOU MADE. PLEASE, PLEASE WISELY PLAN AHEAD AND ASSUME THAT YOU WILL (and others too) FALL BECAUSE ONE PERSON MUST REMAIN BEHIND FROM THE BRIDGE AND TO OPEN THE DOOR FOR ANYONE WHO DOES NOT HAVE THE CORELLIA CRYSTAL.
Renegade Greys: TOR Guide 