The eight-dimensional adjoint representation of sl3

Given: The Axiom of Choice, $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_3(\mathbb{C})$, its diagonal Cartan $\mathfrak{h}$ of traceless diagonal matrices, the root spaces $\mathbb{C}{E}_{ij}$ of Root systems of the classical complex Lie algebras, the simple roots ${\alpha }_1={\epsilon }_1-{\epsilon }_2$, ${\alpha }_2={\epsilon }_2-{\epsilon }_3$, and the adjoint representation of $\mathfrak{g}$ (Adjoint representation of a Lie algebra).

The Axiom of Choice is assumed; it enters through the root-space and highest-weight theory used below (The Axiom of Choice).

The adjoint representation of ${\mathfrak{s}\mathfrak{l}}_n(\mathbb{C})$ has highest vector $E_{1n}$ and highest weight ${\epsilon }_1-{\epsilon }_n$, the highest root (The adjoint representation and highest root, Highest-weight vectors and modules).

The roots of ${\mathfrak{s}\mathfrak{l}}_3$ are the six functionals ${\epsilon }_i-{\epsilon }_j$ with $i\mathrm{\ne }j$, with one-dimensional root spaces $\mathbb{C}{E}_{ij}$, and $\mathfrak{h}$ has dimension $2$ (Root systems of the classical complex Lie algebras). We choose ${\mathrm{\Phi }}^{+}=\{{\epsilon }_i-{\epsilon }_j:i<j\}$; directly from this three-element set, its indecomposable positive roots are ${\alpha }_1={\epsilon }_1-{\epsilon }_2$ and ${\alpha }_2={\epsilon }_2-{\epsilon }_3$, so they are its base and ${\alpha }_1+{\alpha }_2={\epsilon }_1-{\epsilon }_3$ (Positive systems and simple roots).

The fundamental weights satisfy ${\omega }_k(h_{{\alpha }_j})={\delta }_{kj}$ with $h_{{\alpha }_j}=E_{jj}-E_{j+1,j+1}$; for $k=1,2$ one computes $({\epsilon }_1-{\epsilon }_3)(h_{{\alpha }_1})=1={\omega }_1(h_{{\alpha }_1})+{\omega }_2(h_{{\alpha }_1})$ and $({\epsilon }_1-{\epsilon }_3)(h_{{\alpha }_2})=1={\omega }_1(h_{{\alpha }_2})+{\omega }_2(h_{{\alpha }_2})$, so ${\epsilon }_1-{\epsilon }_3={\omega }_1+{\omega }_2$ (Fundamental weights, Standard and dual representations of sl_n).