I’m sorry but you used so many big words so close together, I could find no purchase with the things I know. Perhaps you could tell me what was the problem that Lie groups are supposed to solve. I read something about an infinitely symmetric transformation- the rotation of a circle is a Lie group. But then what are all the letters about- the A,B,C etc all the way up to E 8. And why did it take 77 hours on the Sage supercomputer to solve E8.