It is possible that you have already been here. Kogoruhn, an abandoned Dwemer Stronghold in the Ashlands southeast of the Falamaryon stronghold, is the destination of a Ghostgate Temple quest to retrieve the Hair Shirt of St. Aralor (it's on the body of an Ordinator behind a big pile of dirt on the southwest side of the main building -- there is also a full set of Indoril armor, if you're of a mind to take it). If you have already completed this quest and thoroughly explored Kogoruhn, you may already have the three items that you need: Corprus Weepings, a House Dagoth cup and the Shadow Shield. If so, simply talk to Sul Matuul again to proceed to the Third Trial (he will tell you to keep the items).
If you have not already been here, you're in for a toughie. This place is swarming with big and little Dagoths.
The first two items are a cinch to retrieve. They are in the Dome of Pollock's Eve outbuilding and guarded by one creature, Dagoth Girer. The cup is to your left as you enter, next to a bowl. There are some Corprus Weepings under a brazier directly behind the central pillar, but any Corprus Weepings will do and you'll have plenty of opportunity to get some as you make your way to the Shield.
The three outbuildings (no propylon chamber) are straightforward.
If you want to do a thorough job, get up to the top of the main building and go in through the roof entrance. If you are just concerned with getting to the Shadow Shield, the shorter route is to use the ground-level entrance.
Starting from the top (the interiors are pretty straightforward, so I'm not going to bother with maps -- kind of hard to get lost in an essentially circular corridor):
You'll also find an Ebony Spear and some repair prongs in this area, which is about the only reason to come here.
There is an Iron Tanto stuck through a note (interesting reading) on a table in the one chamber on this level.
How nice. No Dagoths in this area. But that doesn't mean you won't have company.
They're scattered so you can take them one at a time. There are some rather nice goodies in the area, though. Behind some locked cell doors you will find:
There are two exits from this area: Vault of Aerode and the Nabith Waterway. You want the waterway.
Elsewhere in the area you will find a Telvanni Dust Adept Helm, an Exquisite Robe, a Silver Claymore, Steel Greaves, Standard Potions (Detect Key, Restore Speed, Fortify Strength), 309 gold laid out in a Dagoth symbol.
There is no other exit from this area, so go back the way you came.
This is an area similar to the Underworks of the cantons in Vivec. Your objective is to keep heading south until you get to the door leading to Charma's Breath. There are a few minor treasures in various crates and barrels in the area, plus 3 dead adventurers (a few golds, a random weapon and (maybe) a bonemeal). Aside from a Dwarven Mace laying on top of a barrel and a barrel full of arrows, there is nothing noteworthy. There are, however, a few creatures who will try to stop you from making it to the door.
The objective at this point is not to clear out this area, but simply to get through it. Your shortest route is to take the first right and go through the door to the Bleeding Heart area. There are no treasures laying around, and only one of the creatures has anything worthwhile, but it's going to be a tough fight to get it.
There is an exit door behind Dagoth Uthol that opens into the Red Mountain Region.
A few creatures block your way, but the Shadow Shield is in the chamber at the western edge of this area.
Once you've cleared it out, there are a lot of very nice goodies in the chamber (it's a bit dark, so a Night-Eye spell or potion can do a lot of good in finding them):
On your way out, you are likely to find that some 6th House and Ash Creatures have respawned if you disposed of their bodies on the way in, so teleport out if you can't or don't want to fight your way out.