What is the procedure in a twin jet, if an engine problem is suspected?
In a single engine aircraft, you obviously continue flying under power, to the nearest plausible landing point, regardless of what the engine is doing (short of an engine fire, perhaps).
But in a twin, should one always attempt to idle / shut down one engine (one at a time, obviously) to see if a problem goes away?
I suppose, knowing nothing about these light jets, that the drawback of doing nothing (and treating it like the SE case, and returning to the airport under power) is that if an engine does have a real fault and disintegrates in an uncontained manner and shreds the control cables, you lose the tail control surfaces which is obviously very bad news, or maybe it could damage/ignite fuel lines. Otherwise, why would anybody want to shut down an engine at all?
I also do not understand the comments about the LHS being licensed to fly the aircraft alone. Judging from the passengers, this was "obviously" a charter flight, and all charters need 2 pilots for a jet. Was it not a charter flight, or is there some other factor there? What if the passengers did not pay for the flight but were carried for free?