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In the crystallographic convention, a Coxeter graph is admissible when it is realized by linearly independent unit vectors $v_i$ in a Euclidean space, with nonpositive mutual inner products and $4(v_i,v_j{)}^2=n_{ij}\in \{0,1,2,3\}$ for distinct vertices. The number $n_{ij}$ records a single, double or triple connection; equivalently, the displayed Gram matrix is positive-definite. In edge-label notation the nonzero connections have labels $3,4,6$. Forgetting multiplicity, the underlying graph is a forest: for any cycle of $k\ge 3$ distinct vertices, the sum of the corresponding unit vectors would have squared norm at most $k-k=0$, contradicting their linear independence. Multiple lines on one connection are labels and are not length-two cycles. This is the convention of Wehler's Lie algebra notes, section 6.3.

An irreducible Coxeter system is one whose Coxeter graph is connected. The connected components of a Coxeter graph give the direct-product decomposition of its Coxeter group.

In the finite-dimensional complex semisimple Lie algebra setting implicit in this root-system construction, the given Cartan subalgebra and root spaces determine a reduced crystallographic root system $R$. The ordering divides $R$ into positive roots and their negatives. The simple roots are the positive roots that cannot be written as a sum of two positive roots; a standard root-system fact is that they form a basis and that every positive root is a nonnegative integral combination of them.

Index the simple roots as ${\alpha }_1,\dots ,{\alpha }_{\ell }$ and form the Cartan matrixFor $i\ne j$, these entries are nonpositive integers and $A_{ij}{A}_{ji}\in \{0,1,2,3\}$. Construct one vertex per simple root and join distinct vertices by $A_{ij}{A}_{ji}$ lines. If root lengths differ, direct the arrow toward the shorter simple root. This labelled and oriented graph is the Dynkin diagram. Its unoriented Coxeter graph is admissible: normalizing the simple roots to unit length gives $4(v_i,v_j{)}^2=A_{ij}{A}_{ji}$. A general Lie algebra without the semisimple root-system hypotheses need not produce such a diagram.

In the crystallographic convention for this course, an admissible Coxeter graph comes from linearly independent unit vectors $v_1,\dots ,v_r$ in a Euclidean space, such that for $i\ne j$,The vertices $i,j$ have no connection when $n_{ij}=0$, and a single, double or triple connection when $n_{ij}=1,2,3$. Equivalently, the symmetric Gram matrixmust be positive-definite. This equivalence follows because a positive definite Gram matrix can be realized by linearly independent vectors, while such vectors necessarily have a positive definite Gram matrix. In the edge-labelled Coxeter graph convention, these connections have labels $3,4,6$, since the corresponding inner products are $-\cos (\pi /3)$, $-\cos (\pi /4)$ and $-\cos (\pi /6)$.

If the three simple roots are distinct and every pair of root spaces had nonzero Lie bracket, part (iv) would put all three connections in the Dynkin diagram. Their underlying Coxeter graph would contain a triangle, contradicting part (ii). Hence at least one pair satisfies $\boxed{[{\mathfrak{g}}_{{\alpha }_i},{\mathfrak{g}}_{{\alpha }_j}]=0}$. If two of the listed roots coincide, their common root space is one-dimensional, so its bracket with itself is already zero. Thus the conclusion also covers that case.

A simply-laced positive-definite Coxeter graph has no multiple edges and has positive-definite symmetric Cartan matrix, with diagonal entries $2$ and entry $-1$ precisely across an edge. Its connected components are exactly the ADE classification

The graph underlying $1\to 2$ is $A_2$. For simple roots ${\alpha }_1,{\alpha }_2$ with $({\alpha }_i,{\alpha }_i)=2$ and $({\alpha }_1,{\alpha }_2)=-1$, its roots areThere is one base for each Weyl chamber, hence six bases. The Weyl group is generated by the two root reflections withso $W\simeq S_3$. Relative to a chosen base, the two Coxeter elements are $s_1{s}_2$ and $s_2{s}_1$; both have order three.

A representation of the quiver is one linear map $f:V_1\to V_2$. Choosing bases that put $f$ in rank normal form decomposes it into copies ofThese are indecomposable and have dimension vectors $(1,0)$, $(0,1)$, and $(1,1)$, the three positive roots of $A_2$. More generally, Gabriel theorem says that for any orientation of a simply-laced positive-definite Dynkin graph, indecomposable quiver representations are in bijection with its positive roots. The finite ADE root system therefore gives finitely many indecomposables.

The Coxeter graph has vertex set $I$. Distinct vertices $i,j$ are joined precisely when $m_{ij}\ge 3$, and the edge is labelled $m_{ij}$ when $m_{ij}>3$; the customary unlabelled edge therefore means $m_{ij}=3$. The Coxeter system is an Irreducible Coxeter system precisely when this graph is connected.