operating with the $W(A1)$ one gets the other 2 states of the adjoint representation, $p=0$ and $p=-2$. The edge from $p=-2$ to $p=2$ forms the root polytope which is the dual of the Voronoi polytope, and is represented as $(2)$, which also indicates the adjoint representation.

-   Note that the building block is the Delone polytope which is an edge of lenght 1.

For $A_2$ we present the representations of interest in Figure \[A2polytopes\], and note that the Voronoi cell is the hexagon made of the 2 triangles (10) and (01), while the Delone cell’s building blocks are the triangles whose centers are the vertices of the Voronoi cell. Each one is equivalent to one of the triangles (10) and (01) $-$ and six of them make the hexagon root polytope.

![$A_2$ “polytopes”.[]{data-label="A2polytopes"}](A2polytopes.png)

It is not necessary to introduce the polytopes of the exceptional Lie algebras because we will work with $G_2$ and $E_8$ only as a container for the interaction between its $A_2$ subalgebras, which will be our main objects.

To present a general representation of the roots and the weights of $A_{n}$ we add one vector to our orthonormal set of vectors, $l_{i}$ $(i=1,2,...,n+1)$, $(l_{i},l_{j})=\delta_{ij}$ . We define 