After getting all 4 Aetherium Shards, Katria tells you she will meet you in the Forge

This objective is completed: Search for the Aetherium Shards (4/4)
And you receive the next objective: Locate the Aetherium Forge

Your quest log updates:

Lost to the Ages
In Arkngthamz, I met Katria, a ghost searching for the Mythical Aetherium Forge. With her help, I found the four Aetherium Shards that make up the key to the Forge, and now seek entrance to the Forge itself.

Lost to the Ages Forge Location
Travel to the Ruins of Bthalft. It is located southeast of Ivarstead, close to Treva's Watch. If it is not marked on your map, the easiest way to find it is to go to Treva's Watch (also southeast of Ivarstead) and travel straight west, past the river.

Clear the bandits in the area when you get there, then Speak with Katria. Ask her what to do now. She tells you to look at the device here. The gear in the center is just about the right size. Try putting the shards in and we'll see what happens.

Place the Aetherium Shards on the Dwemer Mechanism. You have to activate it four times to place each piece of the Aetherium Shards. The mechanism starts spinning. Katria suggests you try taking it out now. Retrieve the Aetherium Crest. The ground starts to tremble, so Stand Clear. The platform rises, revealing some sort of elevator tower. This is the entrance to a long-sealed ruin. You now need to Find the Aetherium Forge. Approach the lever and activate to open the Aetherium Forge. You go down into the ruins, Katria mentions it took longer than usual so you are probably very deep below the ground.

Activate another lever to open the gate, then go down a bunch of stairs. Turn west and climb up some stairs, soon arriving at an antechamber with a tree in it. There will be a locked door ahead, with two statues on the sides. Note the two resonators on top. Shoot both of them to unlock the door. Go west down into the corridor, then turn left.

You arrive at the forge, but the fumes coming from it are toxic. You need to Shut off the Steam. There are two Steam Valves on the pipes that need to be turned, climb the platforms on each side and find the red valves, turn them to switch the steam off. Head towards the forge but some enemies appear. You now need to Defeat the Guardians of the Forge. These include Dwarven Spider Guardians, Dwaren Spider Workers, Dwarven Sphere Masters. The steam will be turned back on during the fight, so you have an optional objective of shutting the steams down again.

When you defeat all these enemies, a boss named "The Forgemaster" emerges from the lava behind the forge. Don't use fire based attacks on it, as it is resistant to fire. Try frost based attacks. Kill the Forgemaster. Then, Speak with Katria again. She says there is only one thing left to do. We have to prove this actually works. That this is the real Aetherium Forge. You can prove this by forging something.

You get the following objectives:
(Optional) Search for Crafting Materials
Use the Aetherium Forge

Katria mentions that with the Aetherium Shards you collected, and the materials in this room, something can be forged. She tells you to make it count. This may be the only chance you ever get to use the Forge.

There is a chest nearby that has some crafting materials including Dwarven Metal Ingots, Ebony Ingots, Flawless Ruby, Flawless Sapphires, Gold Ingots, Refined Malachites, and Silver Ingots. Loot the chest to complete the optional objective. Another chest will have more Gold Ingots and other needed materials in it.

Use the Aetherium Forge. You get an option of choosing to create one of the following items. You can only make ONE item and you will NOT be able to use the forge again, so choose carefully. If you cannot make the item you want, search the Forge room for additional crafting materials.

The items are:

Aetherial Crown
Retains the last Standing Stone ability you held, granting you its effects in addition to those of your current Stone.

Aetherial Shield
Enemies struck by this shield become ethereal for 15 seconds, making them unable to attack or be attacked

Aetherial Staff
Summon a Dwarven Spider or Sphere for 60 seconds, wherever the caster is pointing.

Which Aetherial Item to Create?
Choose the item to create. The best Aetherial Forge item in my opinion is the Aetherial Crown. But choose the item that appeals to you the most.

Katria thanks you and bows before you.

This completes the quest, "Lost to the Ages", congratulations! You also unlock the Lost to the Ages Achievement.