What is the simplest way to derive an $E_8$ algebra? I am not interested in $E_8$ itself but what would compel one to think about it. I know for example why you would want to think about $SU(2)$ and it makes sense. The reason to me is that you can think of rotations, think of other ways of representing rotations, and then you will discover an equation which you can solve and then discover simple spinors. What is the natural way to arrive there for $E_8$?
I know most of you might say, take a graduate course or read a book, but I seriously think that there should be an overview-level to understand why the $E_8$ algebra is an interesting object, from a layman's perspective.