When, as a young man, I attended many concerts by the Greats of the keyboard, I was a fan of Rubinstein, whom I had the fortune to know personally and to be able to speak with several times. One could never find a seat in the front rows, but at the end of the concert, during the encores, the audience could approach the stage to see the great pianist playing up close. I remember a May 1st at the Aula Magna of the University of Rome; the audience had invaded the stage area and was around him, as he did not stop playing. In particular, I remained curious about how he executed the thumb passage. This could be clearly seen when the stage was high and I found myself in such a position as to observe the hands from below. He simply let the thumb "slide out" from the keyboard to allow the 4th or 3rd finger to continue the succession without "climbing over." A completely natural movement that did not force his hand to "reorient" itself in one direction or another along the keyboard. He performed what is called "displacement," which excludes the thumb passing "underneath" or the fingers passing "over."
Now, one must learn to govern the hand in its balance both in scales and arpeggios, where this execution is a bit more difficult. I have already spoken about this much in many posts, and I give an example in the Tutorial dedicated to Mozart's Sonata K330, where several types of scales are present.
The study of scales must be primarily "mental." Hundreds of books on scales have been printed, as if every time something "different" could be said. Incredible!
In reality, study stems from the knowledge of keys and the awareness of the fingering to be used.
So, sit down in front of the keyboard, think about the scale, its alterations, and try to visualize the fingering to be used. Start with scales in descending motion for two octaves. Decide whether to stop after one octave or to perform two in succession.
Then, when we are ready, take a deep breath and EXECUTE THE SCALE WITHOUT ANY INTERRUPTION SHOULD... AN EARTHQUAKE OCCUR!
Only at the end will we evaluate the errors and the reason for them. Correction and re-execution.
Try. It is not easy.
P.S.: obviously, we will move on to four octaves and ascending motion as soon as... we have clear ideas.
Happy studying