= Solution
A <Dynkin diagram> has one vertex for each <simple root>. Vertices $i,j$ are joined by $a_{ij}a_{ji}$ bonds, where $a_{ij}$ are the off-diagonal <Cartan integers>. No bond means orthogonality; one, two and three bonds mean angles $2\pi/3,3\pi/4,5\pi/6$. A multiple-bond arrow points to the shorter root. Thus the <Dynkin diagram> determines the <Cartan matrix>, including the relative lengths inside each connected component.
An <irreducible root system> corresponds to a connected <Dynkin diagram>. The <classification of finite crystallographic Dynkin diagrams> gives
$$
\boxed{A_l\ (l\ge1),\ B_l\ (l\ge2),\ C_l\ (l\ge3),\ D_l\ (l\ge4),\ E_6,E_7,E_8,F_4,G_2.}
$$
Here $C_2$ is already represented by $B_2$, and $B_1,C_1$ by $A_1$. General systems are orthogonal direct sums of these, with separate component scales. The following diagrams specify all the types; dots denote a continued chain, not additional vertices.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-4-dynkin-diagrams.png]
{title=Finite irreducible crystallographic Dynkin diagrams, with every arrow pointing toward the shorter root}
For orientation, the <An Dynkin diagram> is a single chain, the <Bn Dynkin diagram> has a terminal double bond pointing to its terminal short root, and the <Cn Dynkin diagram> reverses that arrow. The <Dn Dynkin diagram> has arm lengths $1,1,l-3$. The <En Dynkin diagram> has arm lengths $1,2,2$; $1,2,3$; or $1,2,4$. The <F4 Dynkin diagram> has a central double bond and the <G2 Dynkin diagram> a triple bond.
The positivity behind this finite list is the positive-definite Gram matrix of the <simple roots>. In particular the underlying graph is a tree: if it contained a cycle, the sum of its normalized roots would have squared norm at most zero, since each bonded inner product is at most $-1/2$. This contradicts <linear independence>. Restricting the possible bonds and branches by positive definiteness gives precisely the listed chains and trees. Conversely their <Cartan matrices> are positive definite after symmetrization and their reflection-generated root sets give the indicated finite <root systems>.
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