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See: Math notebook, Classical Lie groups, Classical Lie Root systems?, Intuiting classical root systems
Intuit the five exceptional root systems.
The four classical root systems are as follows, where throughout, {$i>j$}:
| {$A_n$} | {$\pm (x_i-x_j)$} |
| {$B_n$} | {$\pm (x_i-x_j), \pm (x_i+x_j), \pm x_i$} |
| {$C_n$} | {$\pm (x_i-x_j), \pm (x_i+x_j), \pm 2x_i$} |
| {$D_n$} | {$\pm (x_i-x_j), \pm (x_i+x_j)$} |
The five exceptional root systems are described in Wikipedia's article on root systems.
| {$G_2$} | {$ (x_i-x_j), (x_i - x_j) - (x_j - x_k)$} for all distinct {$ i,j,k \in \{1,2,3\}$}. |
Thus {$G_2$} is the disjoint union of two copies of {$A_2$}.
Alternatively,
| {$G_2$} | {$ \pm (x_i-x_j)$} where {$i \neq j$}, and {$\pm (3x_i - (x_1+x_2+x_3))$} for all {$i$}. |
| {$F_4$} | {$\pm (x_i-x_j), \pm (x_i+x_j), \pm x_i, \frac{1}{2}(\pm x_1 \pm x_2 \pm x_3 \pm x_4)$} |
Thus {$F_4$} has a copy of {$B_4$} as a subset.
| {$E_8$} | {$\pm (x_i-x_j), \pm (x_i+x_j)$}, and {$\frac{1}{2}(\sum_{i=1}^{8}(-1)^{a_i}x_i)$} where {$\sum_{i=1}^{8}{a_i} \in 2\mathbb{Z}$}. |
{$E_7$} and {$E_6$} are subsets of {$E_8$}.
Literature
Notes by Lecture 16 on Lie Algebras by Kevin McGerty.