The following page of our guide to Doom Eternal explains how to reach the first Slayer Gate on the Exultia map and obtain the Empyrean key.
How can I open a Slayer Gate in the Exultia level?
The first gate is quite easy to spot, as it is visible once you pass a section where you make a lot of jumps over lava.
As for getting the key, enter the cave next door, as if you were going to continue the mission. Kill the Gargoyles and jump upstairs - turn around now and fly through a recess in the wall. You'll find the key on a levitating piece of the ground, practically above the gate. Jump off and open Slayer's Gate to join the battle.
Slayer Gate Battle 1 - Exultia level
Initially, you'll encounter a massive gargoyle attack, which can be easily defeated with grenades and fast-shooting firearms. A moment later, a Dread Knight and some Revenants will appear. However, this is the easiest of all gates and you should be able to manage it without much trouble. Use chainsaw quite often. Run away from Dread Knight and attack him while he's in air - then you will obtain the first Empyrean key.