Theorem: Every finite irreducible root system corresponds to one of the nine types in the classification.

Strategy: Analyze connected Dynkin diagrams by showing that the associated Cartan matrix must be positive definite, severely constraining the possible diagrams.

Step 1: Positive definiteness requirement

For a finite root system, the Cartan matrix $A = (a_{ij})$ must be positive definite. This follows because $A$ represents the matrix of the Killing form restricted to the Cartan subalgebra in a suitable basis.

Step 2: Simply-laced case (all edges simple)

When all roots have equal length, $a_{ij} \in \{0, -1, 2\}$ and $A$ is symmetric. The diagram has no multiple edges. We must classify connected graphs where the adjacency matrix $-A + 2I$ gives a positive definite $A$.

Subcase: Linear diagrams $A_n$: $\circ - \circ - \cdots - \circ$

The Cartan matrix is tridiagonal with 2's on diagonal and -1's on off-diagonals: $A_{A_n} = \begin{pmatrix} 2 & -1 & & \\ -1 & 2 & \ddots & \\ & \ddots & \ddots & -1 \\ & & -1 & 2 \end{pmatrix}$ This is positive definite for all $n \geq 1$ (eigenvalues are $2 - 2\cos\frac{k\pi}{n+1}$ for $k = 1, \ldots, n$).

Subcase: Diagrams with one branch point $D_n$: The fork diagram is positive definite for $n \geq 4$ but not for $n = 3$ (which gives affine type).

Subcase: Exceptional simply-laced $E_6, E_7, E_8$: Extended $A_n$ diagrams with one additional node attached. Detailed eigenvalue analysis shows:

• $E_6$: Smallest eigenvalue $> 0$ (positive definite)
• $E_7$: Smallest eigenvalue $> 0$ (positive definite)
• $E_8$: Smallest eigenvalue $> 0$ (positive definite)
• $E_9$ would-be diagram: Smallest eigenvalue $= 0$ (affine, infinite-dimensional)

Thus simply-laced types are exactly $A_n, D_n, E_6, E_7, E_8$.

Step 3: Non-simply-laced cases

When roots have two lengths, we have double or triple edges.

Double edges ($mn = 2$):

• $B_n$: Long-short-short-...-short with arrow
• $C_n$: Short-short-...-short-long with arrow (dual to $B_n$)
• $F_4$: Short-short-long-long arrangement

Verification shows these give positive definite Cartan matrices.

Triple edge ($mn = 3$): Only possibility is $G_2$: Two nodes with triple bond. Any larger diagram would fail positive definiteness.

Step 4: Exclusion of other possibilities

Diagrams with loops, multiple branches, or larger multiplicities all give non-positive-definite Cartan matrices, hence correspond to infinite-dimensional algebras (affine or hyperbolic).

Conclusion: The classification list is complete and exhaustive. □