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Next to the smelter in the outpost is a door to Botanist's Oasis. Within lives the Argonian Botanist Shil'ara, who needs some supplies.
Bring 3 Siligonder Eggs and 5 Pieces of Siligonder Chitin to Shil'ara.
Go back to the outpost area where there should be plenty of Siligonders available. Killing 2-3 should suffice for the supplies. Bring them to Shil'ara in the Oasis.
Go and see if Jelikirr is ready with your new armor.
Jelikirr is back in the supply post usually near the smelter. By the time you go outside he should have your armor ready.