Well for starters you only have one army.
An army is either unbound or battle forged. I'm just gonna ignore unbound for now.
A battleforged army consists of any number of detachments and/or formations.
So a black crusade is a detachment. In parallel to that you could have a Cyclopia Cabal formation.
An allied detachment is a different kind of detachment. That would also exist in parallel to the crusade and cabal. Don't let the name confuse you, it's just a normal detachment. It has nothing to do with the allied matrix per se.
Some of those come with additional restrictions though, like the allied detachment. It has to be a different faction than whatever you designate your primary detachment, but you can still have as many as you want.
The allied matrix is of no concern to detachments and such. Those limitations only come into play on the tabletop, so you just need to be aware that the matrix might make life a bit difficult.
So yes, you could have an army consisting of a crusade, a cabal, and an allied KDK detachment.
Oh and you can apply the Black legion rules to the crusade detachment, but you can't benefit from the free VotLW for the cabal because that isn't part of the crusade detachment.
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