[proofplan] We pass from an irreducible reduced root system to its base of simple roots and hence to a connected Dynkin diagram. The structural inputs are the existence of simple-root bases, the rank-two classification of reduced root systems, the finite Dynkin diagram classification theorem for positive-definite symmetrizable Cartan matrices, and the uniqueness theorem saying that a finite crystallographic root system is determined up to isomorphism by its Cartan matrix. After verifying the hypotheses of these results, the low-rank coincidences explain the stated index ranges, giving existence and uniqueness of exactly one listed type. [/proofplan] [step:Choose simple roots and attach the connected Dynkin diagram] Let $\Phi \subset E$ be an irreducible reduced root system in a finite-dimensional Euclidean [vector space](/page/Vector%20Space) $E$ with inner product $(\cdot,\cdot)_E$. Since $\Phi$ is finite, reduced, and crystallographic, the existence theorem for simple-root bases of finite root systems applies. Choose a base of simple roots \begin{align*} \Delta = \{\alpha_1,\dots,\alpha_n\} \subset \Phi, \end{align*} where every root is an integer linear combination of the $\alpha_i$ with all coefficients either non-negative or non-positive. Define the Cartan integers \begin{align*} a_{ij} := \frac{2(\alpha_i,\alpha_j)_E}{(\alpha_j,\alpha_j)_E}\quad \text{for }1 \le i,j \le n. \end{align*} The Cartan matrix $A=(a_{ij})$ satisfies $a_{ii}=2$, $a_{ij}\in \mathbb Z_{\le 0}$ for $i\ne j$, and $a_{ij}=0$ iff $a_{ji}=0$. Attach to $\Delta$ the Dynkin diagram $\Gamma(\Phi,\Delta)$ with vertices $1,\dots,n$, with $a_{ij}a_{ji}$ edges between $i$ and $j$, and with the arrow, when $|\alpha_i|\ne |\alpha_j|$, pointing toward the shorter root. Since $\Phi$ is irreducible, $\Delta$ cannot split into two mutually orthogonal non-empty subsets. Therefore $\Gamma(\Phi,\Delta)$ is connected. [/step] [step:Restrict the possible edges using rank-two root subsystems] For $i\ne j$, let $E_{ij}:=\operatorname{span}_{\mathbb R}\{\alpha_i,\alpha_j\}$ and let \begin{align*} \Phi_{ij}:=\Phi \cap E_{ij} \end{align*} be the set of roots lying in this two-dimensional subspace. Since $\Phi$ is reduced and crystallographic, $\Phi_{ij}$ is a reduced crystallographic root subsystem of $E_{ij}$; its simple roots are $\alpha_i$ and $\alpha_j$ after restricting to the positive system determined by $\Delta$. The rank-two classification theorem for reduced crystallographic root systems therefore applies and gives only the types $A_1\times A_1$, $A_2$, $B_2=C_2$, and $G_2$. For this subsystem the Cartan integers attached to the simple pair are still \begin{align*} a_{ij}=\frac{2(\alpha_i,\alpha_j)_E}{(\alpha_j,\alpha_j)_E},\qquad a_{ji}=\frac{2(\alpha_j,\alpha_i)_E}{(\alpha_i,\alpha_i)_E}, \end{align*} so the rank-two classification gives \begin{align*} a_{ij}a_{ji}\in \{0,1,2,3\}. \end{align*} The cases $0,1,2,3$ correspond respectively to $A_1\times A_1$, $A_2$, $B_2=C_2$, and $G_2$; equivalently, the non-orthogonal angles are $2\pi/3$, $3\pi/4$, and $5\pi/6$, with root-length squared ratios $1$, $2$, and $3$ in the multiply-laced cases. Thus every edge in $\Gamma(\Phi,\Delta)$ is a single, double, or triple edge, and no other local configuration can occur. [/step] [step:Use positive definiteness to classify the connected diagrams] Let $D$ be the diagonal matrix with entries \begin{align*} d_i := \frac{(\alpha_i,\alpha_i)_E}{2}\quad \text{for }1\le i\le n. \end{align*} Then $G:=AD$ has entries $G_{ij}=(\alpha_i,\alpha_j)_E$, so $G$ is the Gram matrix of the basis $\Delta$. Since the Euclidean inner product on $E$ is positive definite and $\Delta$ is linearly independent, $G$ is positive definite. Hence $A$ is symmetrizable positive definite. Define the quadratic form attached to the diagram by \begin{align*} Q: \mathbb R^n &\to \mathbb R,\\ (x_1,\dots,x_n) &\mapsto \sum_{i,j=1}^n G_{ij}x_i x_j. \end{align*} The preceding paragraph proves that $Q(x)>0$ for every non-zero $x\in \mathbb R^n$. The finite Dynkin diagram classification theorem applies to a connected symmetrizable generalized Cartan matrix whose symmetrization is positive definite and whose off-diagonal products satisfy $a_{ij}a_{ji}\in\{0,1,2,3\}$. Its hypotheses are exactly the connectedness from irreducibility, the Cartan-integrality and sign conditions from the simple-root construction, the rank-two restriction from the previous step, and the positive definiteness of $Q$. The theorem classifies the resulting connected diagrams as precisely \begin{align*} A_n\ (n\ge 1),\quad B_n\ (n\ge 2),\quad C_n\ (n\ge 3),\quad D_n\ (n\ge 4),\quad E_6,E_7,E_8,F_4,G_2. \end{align*} In the standard proof of this finite-diagram theorem, the same form $Q$ is evaluated on explicit integer vectors supported on forbidden subdiagrams: cycles give a non-zero vector with $Q\le 0$, vertices with excessive branching give a non-zero vector with $Q\le 0$, and the inverse Cartan matrices of the simply-laced arms give the branch-length bound that leaves only $D_n,E_6,E_7,E_8$ beyond the paths $A_n$. The multiply-laced case is then reduced by the rank-two products $1,2,3$ and the positivity of $Q$, leaving only the chains $B_n,C_n$ and the exceptional diagrams $F_4,G_2$. Therefore $\Gamma(\Phi,\Delta)$ is one of the diagrams in the displayed list. [/step] [step:Reconstruct the root system from its Cartan matrix] For each finite Dynkin diagram in the list, construct the corresponding reduced root system in the standard Euclidean model: $A_n$ in the hyperplane of $\mathbb R^{n+1}$ with coordinate sum zero, $B_n$ and $C_n$ in $\mathbb R^n$, $D_n$ in $\mathbb R^n$, and the exceptional systems $E_6,E_7,E_8,F_4,G_2$ in their standard lattice models. In each case the displayed simple roots have exactly the Cartan matrix encoded by the diagram, so every listed diagram is realized by a reduced irreducible root system. Conversely, use the uniqueness theorem for finite crystallographic root systems with a fixed Cartan matrix: if $\Phi$ and $\Psi$ have ordered bases $\Delta_\Phi=\{\alpha_1,\dots,\alpha_n\}$ and $\Delta_\Psi=\{\beta_1,\dots,\beta_n\}$ with the same Cartan matrix, then the [linear map](/page/Linear%20Map) \begin{align*} T: \operatorname{span}_{\mathbb R}(\Delta_\Phi) &\to \operatorname{span}_{\mathbb R}(\Delta_\Psi),\\ \alpha_i &\mapsto \beta_i \end{align*} induces a bijection $T(\Phi)=\Psi$. The theorem applies because both systems are finite, reduced, and crystallographic, and the ordered bases have identical Cartan integers. Equivalently, the simple reflections determined by the rows of the Cartan matrix act on the same root lattice $\mathbb Z^n$, and the finite Weyl-group orbit of the simple roots gives exactly the same set of coefficient vectors in both systems. Hence $T$ sends roots to roots bijectively and is an isomorphism of root systems. Therefore the Dynkin diagram determines the irreducible reduced root system up to isomorphism. [/step] [step:Exclude duplicate names and conclude uniqueness] The low-rank coincidences are accounted for by the ranges in the statement: $B_1=C_1=A_1$, $D_2=A_1\times A_1$ is reducible, $D_3=A_3$, and $B_2=C_2$. Thus the list deliberately starts with $B_n$ for $n\ge 2$, $C_n$ for $n\ge 3$, and $D_n$ for $n\ge 4$. Combining the previous steps, every irreducible reduced root system has a connected finite Dynkin diagram from the stated list; each diagram is realized by a reduced irreducible root system; and the diagram determines the root system up to isomorphism. After the low-rank identifications are removed by the stated index ranges, the type is unique. This proves that every irreducible reduced root system is isomorphic to exactly one of the listed types. [/step]