Yes, but as I read the 4 tetrads, some attention remains with the breath throughout the practice, so full attention cannot be given to any of the four frames.
4 tetrads belong to Vipassana, because by looking into the behaviour of 4 tetrads nakedly (without involving concept), you will slowly and passively understand the true nature of 4 tetrads.
If you want to fully understand something, of course you should put full attention to it. We can't do half looking at the breathing, half looking at one of four tetrads.
When you are ready to move to 4 tetrads, do you still need to look for breathing? No.
Breathing is just a technique. A technique to make you focus without any sway. Because it is just a technique to make you focus, you can replace that breathing with anything you like, like a blue flower, like a statue of buddha, like a stone. Anything.
But, once you have gain your unsway focus, if you never move from that point, and just following your breathing, you will not make any single progress in wisdom.
You will make a progress in concentration aspect, but not a single aspect in your wisdom aspect.
Once your mindfulness has been sharp and natural, leave your breathing behind. At this point, your breathing is already hardly noticeable.
At that point, if you keep searching for breathing, you have waste your time. Because when you look for breathing, your "unnoticeable" breath will turn to gross breath. This is not good.The grosser your breathing, the more agitate your mind with concept.
The more unnoticeable your breathing, the clearer your mind (free with concept).
That is the undisputeable basic reality in any pranayama.
Prove it yourself, by holding your breath for 1 minute. Notice your thought. Does it move?
This experiment should give you a confident that the relationship between breathing and movement of thought is one to one.
A very advanced technique to clear your mind is actually holding your breath. However, since this is not taught in Satipatthana, you should rely on the pace of your natural breathing.
Your natural breathing will bring you to the state where your breathing is hardly noticeable.
So, it is very important here, once you achieve that, you should notice your mindfulness. Your mindfulness should be relatively sharp here.
Once you can feel your mind is relatively free from flying concepts, you are ready for 4 tetrads.
The most easiest to understand its true nature is the first one - body. Because body is very gross.
The most difficult to understand its true nature is the last one - mind. Because it is the most subtle one.
In all tetrads that you do, you should slowly notice passively without any concept to get a throughout insight that:
1. In this body, there is no body.
2. In this perception, there is no perception.
3. In this feeling, there is no feeling.
4. In this mind, there is no mind.
4 tetrads actually is equivalent to 2 tetrads. THe first one is body, the rest are the different aspect of mind (perception, feelings, mental formations, etc. - all of them are just the aspect of mind)
When you do the 4 tetrad, you no longer put any single attention to the breathing. If you put it there, you are distracting yourself.
At the end of your 4 tetrad, you should slowly experience directly and nakedly what it means by FALSE APPEARANCES - Appear as if it is something, but actually that something is never truly there.
Ask yourself this question:
Can I say something which look like there, but it is not there as something?
Ordinary people see nothing, but madman see something. That is why he is mad.
Buddha see nothing, but ordinary people see something.
Four tetrads will give you that answer through experience.