By the beautiful classification theorem of complex semisimple Lie algebras, we know that there are exactly $5$ types of exceptional Lie algebras, say type $E_6,E_7,E_8,F_4$ and $G_2$. We have a general approach to construct Lie algebras from the corresponding root systems. Howerer, I want to know some concrete and specific constructions. How did Killing and Dynkin do this? I've heard that it's related to Octonions $\mathbb{O}$.
Moreover, I know that exceptional Lie groups and algebras arise as symmetries for many modern physics structures in quantum field theory and string theory, are there any good references?
My background: I know standard theory of complex semisimple Lie algebras and differential geometry. Also I'm familiar with classical mechanics, and know some quantum mechanics. I can read Chinese and English, and both technical and popular literatures are welcome.
Thanks in advance.
Best Answer
There is a nice article by Alberto Elduque on Tits construction of the exceptional simple Lie algebras. Tit's found this construction $1966$, when Dynkin was $42$ years old.
On this site, there are several references, too. For $E_8$ I wrote some references in this post
Exceptional Lie algebras E8
For $G_2$ see also here:
Understanding $G_2$ as a particular subgroup of $SO(7)$
Also, the pages of John Baez here contain interesting informations. On the Lie group level, see also this MO-post:
Beautiful descriptions of exceptional groups