This will pertain to breathing and maintaining an open airway.
I will start out with a concept that several may disagree with. All I ask is that you consider what I'm telling you.
The diaphragm is called an involuntary muscle. It works without us thinking about it . It works when we are asleep. It can MAKE us sneeze or cough. We can however, exert some control over it. We CAN hold our breath , take a breath when we want, take a short gasp or a long deep breath. This indicates a measure of control. In as much as trumpet playing IS AIR and breath control then working on this major source of our breath is vital. There are several Yoga exercises that are excellent as is timed breathing while walking or jogging.
The airway must always be open both in inhaling and in playing. One problem is posture. I've seen many experienced players slumped over while jamming. I've seen them with their heads down our their arms against their ribcage. If we give this its proper importance then we see that these things WILL lead to a closed throat, shallow breaths and poor support.
If the jaw is pushed forward slightly this will cause the throat oppening to be larger than it normally is. Try it move the jaw forward slowly and check if you can feel your throat open up. Think of the effect that can have on your tone. The more forward jaw position will also make your lower lip take on more of the workload. This increases endurance ( after you get used to it). Notice that I said more forward Stevens demanded an even tooth alignment. I advocate moving it until the throat opens. This will be different for every player.
Another key feature in maintaining an open airway is a pivot. You could write hundreds of pages about this. But that's already been done. In a nutshell by raising or lowering the bell of your horn while you are playing you can maintain a more open airway and clearer tone. As you play higher and lower notes the air stream will slightly move in the mouthpiece. If we can keep it lined up with the throat hole the sound is better. The SLIGHT bell movement will produce an opposite movement or realignment of our lips to the mouthpiece. Now which way do you move the bell?
Try this test . Play a low g 1-3.
Move the bell up then move it down. One way should improve the sound. When you move to a lower note from now on always pivot this direction. The opposite direction will aid the upper notes. This is a good movement whenever you have to leap between notes.
The tongue arch has been used for years to speed up the air inorder to play higher notes. Most people arch to the point where the sound quality is affected. Instead of arching up to eeee try aaaaa. This is a more open sound yet it still compresses the air slightly. After all the tongue arch cannot give you an extra octave. It is merely used for rapid note movement. The abdominals compress the air for your range.
Timed breathing is another aspect of playing. Some people always take a deep full breath. When playing in the upper register this creates tension. The upper register takes air compression and speed but not air mass. The low notes need the full breaths. Try a half or quarter breath before you play your next high g. This will allow your muscles to do their job.
As for the Stevens grip on the lower portion of the valve casing. Do it if you want to. But please have your horn adjusted so that you can use your slides properly. And always play with the same grip. This can really confuse your embouchure by switching back and forth.