This page of our guide to Doom Eternal explains how to open the fourth Slayer Gate on the Arc Complex map and obtain the next Empyrean key.
How can I open Slayer Gate in the Arc Complex level?
The key is very close to the gate. Jump on the wall just in front of the gate, then jump towards the key. Return to the gate to enter another battle.
Slayer Gate Battle 4 - Arc Complex level
The battle begins really hard, because a Cyberdemon (a very powerful foe) will appear. This enemy is slow but has tremendous attacking power - move all the time and shoot at it from the sides. If you have a Crucible Blade available, then you can kill the enemy with one blow.
In addition to Cyberdemon, some other monsters will appear like Mancubus or weaker enemies that can be used to replenish health. The next wave are Barons of Hell and Doom Hunter. Both enemies are really strong, especially Doom Hunter. Attack his flying pod. After you win, you will get another Empyrean key.