Airspeed, wing shape, surface area, air density, and—what I'm sure you're alluding to—AoA (adjusted with elevator).
There's a lot of things that are required to facilitate the turn, but it's the horizontal component of lift, generated by the wing, that turns the airplane.
What if the pilot chooses not to turn? How is the air show pilot able to perform a knife-edge pass while remaining on heading, since in knife-edge, every bit of that lift from the wing would then be a horizontal component? (Hint: it has nothing to do with the rudder)
Divorcing the elevator from the lift generated by the wing means the above question cannot be answered properly. On the other hand, recognizing that elevator and lift are indelibly linked, and that it is the pilot's selection of elevator position that controls AoA/lift, makes for a consistent and straightforward explanation.
The explanation of knife-edge flight and a whole host of other possibilities is simple with elevator: in knife-edge flight, the pilot, using the elevator, must drive AoA to the zero lift angle in order to eliminate the lift from the wing. This is the only way to stay on heading in knife-edge flight.
What is true in the knife-edge example in terms of using the elevator to manage lift is true elsewhere. For example: Assume a coordinated, level, upright turn at 60 degrees of bank with sufficient energy/margin to critical AoA.
Q: What is required to sustain such a turn?
A: +2 G
Q: Where does the +2 G come from?
A: Pulling on the elevator ("2 Gs worth" of pull)
Q: What is happening regarding the lift on the wing as a result of the +2 G pull?
A: Due to the +2 G pull on the elevator, total lift doubles; the vertical component, Lv, grows just enough to equal W; the horizontal component, Lh grows as well, accelerating the turn.
Q: What happens if the pilot only pulls +1.5 G?
Q: What happens if the pilot pulls +3 G?
Q: Can you map it on a V-G diagram (or V-n, if you prefer "n" vs. "G")?
Q: What other maneuvers can be mapped on a V-G diagram?
Q: The V-G diagram is a plot of speed (V), G-load (G or n), structural design limits (bending or breaking the airplane), and aerodynamic limits (critical AoA). So, what does the diagram really represent?
A: The V-G diagram a graphic representation of everything the elevator does. You can plot turns, loops, hammerheads, and even accelerated stall scenarios -- all driven by the elevator.