Recovering the Pieces

The mobs that drop these are elite, and they hurt. Bring friends, a tank spec, or a warframe.

Completing both this and The Power to Reforge leads to Forging a Key to the Past.

[item=152204] dropped for me by trash in [zone=8910].

1. 
   
2. The Power to Reforge (250 rep)
   Recovering the Pieces Elite mobs (250 rep)
3. 
   
4. Forging a Key to the Past (25 rep)
5. 
   
6. The Seat of the Triumvirate: Armor of the Triumvirate (350 rep)
7. 
   

Rewards one of these ensembles, depending on your class:

• 
  
• Plate: [item=152244] ([transmog-set=2583])
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• Chainmail: [item=152243] ([transmog-set=2581])
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• Leather: [item=152242] ([transmog-set=2578])
• 
  
• Cloth: [item=152241] ([transmog-set=2577])
• 
  

I can confirm you can easily get the 10 required items from inside the The Seat of the Triumvirate dungeon.

However a following quest after this will need you to go into the same dungeon to complete the quest chain. Meaning you will have to wait for the next day to compete it.

You can't enter the dungeon lockout you completed to do it (it's fairly difficult to solo) because as per most dungeon quests you also have to kill the final boss.