# How to derive an $E_8$ algebra?

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.

• Your intent - the motivation for constructing $E_8$ in the first place - isn't bad, but your wording is. Care to tone down the post and make it sensible? I would do it myself but the revision would be too large to implement without authorisation from the OP, i.e. you. Nov 10 '17 at 21:03
• Bro, I am sorry if my tone sucks. I can learn to fix it. Can you help me make it sound more polite etc?
– user151266
Nov 10 '17 at 21:05
• @JamalS authorization granted
– user151266
Nov 10 '17 at 21:06
• en.wikipedia.org/wiki/… Nov 10 '17 at 21:16
• @JamalS, Oh I see it now :^)
– user151266
Aug 7 '18 at 1:42

There's no reason to think about $E_8$. At least, there's nothing intrinsic to $E_8$ that would make it somehow more appealing over some of the other groups/algebras appearing in the classification of simple Lie algebras. It's large and can accomodate the Standard Model gauge group as a subgroup, but so can many other groups.
The appearance of $E_8$ in theoretical physics is an intriguing technical point - there are no a priori reasons to expect it to show up, but when one does heterotic string theory, it suddenly appears. One way to say why it's special is to say that it is one of the only two groups that can appear as the gauge group of a ten-dimensional SUGRA theory (the low-energy effective limit of a string theory) without causing a gauge-gravitational anomaly that makes the theory ill-behaved quantum-mechanically. For more on the appearance of $E_8$ in heterotic string theory, see this answer of mine and the references therein.