$A_2$, and $E_8$, whose importance for coupling the charge space with spin foam was pointed out in [@Aschheim2], and which contains the other exceptional Lie algebras as well the $A_n$ for lower $n$.

Let us consider a Euclidean space $V$ of dimension $n$. One can introduce an orthonormal basis, $l_{i}$ $(i=1,2,...,n)$, in $V$ so that the Euclidean scalar product $(u,v)$ of $u=\sum_{i}\lambda_{i}l_{i}$ and $v=\sum_{i}\mu_{i}l_{i}$ is $(u,v)=\sum_{i}\lambda_{i}\mu_{i}$. The Coxeter-Dynkin diagrams of $A_n$, $D_n$ and $E_8$ are shown in Figure \[ADE8\_Dynkin\_Diagrams\].

![Coxeter-Dynkin diagrams of $A_n$, $D_n$ and $E_8$.[]{data-label="ADE8_Dynkin_Diagrams"}](ADE8_Dynkin_Diagrams.png)

The nodes on the Coxeter-Dynkin diagram represent the simple roots $\alpha_i$, $(i=1,2,...,n)$, of the associated Lie algebra of rank $n$[^6] and two dots linked by an edge correspond to two roots whose scalar product is $-1$. The other pairs of nodes, which are not connected, correspond to vectors that are orthogonal. The norms are given by $(\alpha_i,\alpha_i)=2$ and the Cartan matrix elements are $$C_{ij}=\frac{2(\alpha_{i},\alpha_{j})}{(\alpha_{j},\alpha_{j})}.\label{eq:cartanmatrix}$$ Complementary to the simple roots, there are the fundamental weight vectors $\omega_{i}$, which are defined by the relation with the simple roots $(\omega_{i},\alpha_{j})=\delta_{ij}$ and are related to each other by $(\omega_{i},\omega_{j})=(C^{-1})_{ij}$. The root lattice $\Lambda$ is the set of vectors $p=\sum_i^n b_{i}\alpha_{i}$, $b_{i}\in \mathbb{Z}$. Similarly the weight lattice $\Lambda^{*}$ is 