classification of indecomposable root systems

There are four infinite families of indecomposable root systems :

$A_n$

$B_n$

$C_n$

$D_n$

The subscript on the name of the root system is the dimension of $𝐄$,the ambient Euclidean space containing the root system. In the case of$A_n$, the ambient $𝐄$ is the $n$-dimensional subspace perpendicularto ${\sum }_{i=1}^n{e}_i$. In the other 3 cases, $𝐄={ℝ}^n$.Throughout, we endow ${ℝ}^n$ with the standard Euclidean innerproduct, and let $e_i,\mathrm{\hspace{0.33em}1}\le i\le n$ denote the standard basis.

As well, there are 5 exceptional, crystallographic root systems:

$G_2$$=A_3\cup \{\pm {\displaystyle \frac{1}{3}}(2e_1-e_2-e_3),\pm {\displaystyle \frac{1}{3}}(-e_1+2e_2-e_3),\pm {\displaystyle \frac{1}{3}}(-e_1-e_2+2e_3)\};$

$F_4$$=B_4\cup \left\{{\displaystyle \frac{1}{2}}(\pm e_1\pm e_2\pm e_3\pm e_4)\right\};$

$E_6$$=A_6\cup \{\pm (e_7-e_8)\}\cup \{{\displaystyle \frac{1}{2}}({\displaystyle \sum _{i=1}^6}(\pm e_i)\pm (e_7-e_8)):\text{4 minus signs}\};$

$E_7$$=A_8\cup \{{\displaystyle \sum _{i=1}^8}(\pm e_i):\text{4 minus signs}\};$

$E_8$$=D_8\cup \{{\displaystyle \sum _{i=1}^8}(\pm e_i):\text{even number of minus signs}\}.$

The following table indicates the cardinality of and the Lie algebrasand Dynkin diagrams corresponding to the above root systems.

Figure 1: Irreducible root systems and simple Lie algebras.