[proofplan]
Attach to the root system its Cartan matrix and Dynkin diagram. Irreducibility is exactly connectedness of this diagram, reducedness gives the usual crystallographic Cartan integers, and finiteness gives positive-definiteness of the associated symmetric form. The classification of connected positive-definite crystallographic Dynkin diagrams gives precisely the displayed list, and the realization-uniqueness theorem for Cartan matrices identifies each such diagram with exactly one isomorphism class of reduced crystallographic finite root systems.
[/proofplan]
[step:Pass from the root system to a connected positive-definite Dynkin diagram]
Let $\Phi \subset V$ be an irreducible reduced crystallographic finite root system in a finite-dimensional real [inner product space](/page/Inner%20Product%20Space) $(V, (\cdot, \cdot))$. Choose a base of simple roots $\Delta = \{\alpha_1, \dots, \alpha_r\} \subset \Phi$, where $r = \dim \operatorname{span}_{\mathbb{R}}(\Phi)$, and define the Cartan integers
\begin{align*}
a_{ij} = \frac{2(\alpha_i, \alpha_j)}{(\alpha_j, \alpha_j)} \in \mathbb{Z}
\end{align*}
for $1 \le i,j \le r$. The matrix $A = (a_{ij})$ is the Cartan matrix of $\Phi$, and the Dynkin diagram $\Gamma(\Phi)$ is the graph with vertices $1,\dots,r$ and edge data determined by the products $a_{ij}a_{ji}$ and the relative root lengths.
The crystallographic hypothesis gives $a_{ij} \in \mathbb{Z}$, reducedness gives $a_{ii}=2$ and excludes non-reduced double roots, and finiteness gives a positive-definite symmetric realization through the inner product on $\operatorname{span}_{\mathbb{R}}(\Phi)$. If $\Gamma(\Phi)$ were disconnected, then $\Delta$ would split into two nonempty mutually orthogonal subsets, and the root system would decompose as an orthogonal direct sum of the corresponding root subsystems. This contradicts irreducibility. Hence $\Gamma(\Phi)$ is a connected positive-definite crystallographic Dynkin diagram.
[guided]
The purpose of this step is to convert the geometric object $\Phi$ into a finite combinatorial object. We start with a base of simple roots $\Delta = \{\alpha_1, \dots, \alpha_r\}$, where $r$ is the rank of the root system, and record all pairwise angle and length information in the Cartan integers
\begin{align*}
a_{ij} = \frac{2(\alpha_i, \alpha_j)}{(\alpha_j, \alpha_j)}.
\end{align*}
Because $\Phi$ is crystallographic, these numbers are integers. Because $\Phi$ is reduced, no root occurs together with twice itself as a root, so the diagonal entries are $a_{ii}=2$ and the usual reduced Dynkin diagram rules apply. Because $\Phi$ is finite in a real inner product space, the form inherited from $(\cdot,\cdot)$ on $\operatorname{span}_{\mathbb{R}}(\Phi)$ is positive-definite.
Now we use irreducibility. If the Dynkin diagram split into two disconnected vertex sets, then every simple root in the first set would be orthogonal to every simple root in the second set. The standard root-string construction then shows that the roots generated by the first set and the roots generated by the second set form two orthogonal root subsystems whose union gives $\Phi$. That is precisely a decomposition of $\Phi$ as a reducible root system, contradicting the hypothesis. Therefore the diagram attached to $\Phi$ is connected.
[/guided]
[/step]
[step:Apply the classification of connected positive-definite crystallographic diagrams]
By the classification theorem for connected positive-definite crystallographic Dynkin diagrams, every connected diagram satisfying the conditions just verified is exactly one of
\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*}
Applying this theorem to $\Gamma(\Phi)$ proves that the Dynkin diagram of $\Phi$ occurs in the displayed list.
[guided]
We have reduced the classification problem to the diagram classification problem. The input to the diagram classification theorem is a connected positive-definite crystallographic Dynkin diagram. The previous step verified each of those hypotheses for $\Gamma(\Phi)$: connectedness came from irreducibility, crystallographicity came from the integrality of the Cartan integers, and positive-definiteness came from the ambient real inner product space. Therefore the classification theorem applies to $\Gamma(\Phi)$ and gives exactly the list
\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*}
This proves exhaustiveness at the level of diagrams.
[/guided]
[/step]
[step:Realize each listed diagram by a reduced finite root system]
For each diagram in the displayed list, let $A$ denote its Cartan matrix. The realization theorem for finite Cartan matrices gives a reduced crystallographic finite root system $\Phi_A$ whose Cartan matrix is $A$. Since each listed diagram is connected, the same connectedness argument from the first step shows that $\Phi_A$ is irreducible. Thus every listed type exists.
[/step]
[step:Use uniqueness from the Cartan matrix to identify isomorphism classes]
Let $\Phi$ and $\Psi$ be irreducible reduced crystallographic finite root systems with the same listed Dynkin diagram. Choose ordered bases of simple roots for $\Phi$ and $\Psi$ compatible with that diagram, and let $A$ be the common Cartan matrix. The uniqueness theorem for root systems with a fixed finite Cartan matrix gives an isomorphism of root systems $\Phi \cong \Psi$ carrying each simple root of $\Phi$ to the corresponding simple root of $\Psi$. Hence each diagram in the list determines one isomorphism class.
If two entries in the displayed list were isomorphic as root systems, their Cartan matrices, and hence their Dynkin diagrams, would be isomorphic. The connected positive-definite crystallographic Dynkin diagrams in the displayed list are pairwise non-isomorphic with the stated ranges of ranks. Therefore the isomorphism class attached to one listed type is distinct from the isomorphism class attached to any other listed type. Combining existence, exhaustiveness, and uniqueness proves the theorem.
[guided]
The final point is that the diagram is not merely an invariant that narrows the possibilities; it determines the root system. Suppose $\Phi$ and $\Psi$ have the same listed Dynkin diagram. After ordering their simple roots according to that common diagram, they have the same Cartan matrix $A$. The uniqueness theorem for reduced finite root systems with a prescribed finite Cartan matrix then gives an isomorphism $\Phi \cong \Psi$ preserving the simple-root data.
This proves uniqueness within a type. To rule out overlap between different types, observe that an isomorphism of root systems preserves Cartan integers after choosing corresponding simple systems, so it preserves the Dynkin diagram. The diagrams
\begin{align*}
A_n,\quad B_n,\quad C_n,\quad D_n,\quad E_6,E_7,E_8,F_4,G_2
\end{align*}
are pairwise non-isomorphic in the displayed parameter ranges. Hence no root system belongs to two different listed types. Together with the realization step and the classification step, this proves that every irreducible reduced crystallographic finite root system is isomorphic to exactly one item in the list.
[/guided]
[/step]
