= Solution
The appropriate abstract object is a finite reduced <crystallographic root system> $R$ in a real <inner-product space> $E$. Its axioms are: $R$ is finite, spans $E$, and does not contain zero; for $\alpha\in R$, $R\cap\mathbb R\alpha=\{\alpha,-\alpha\}$; each <root reflection>
$$
s_\alpha(v)=v-\frac{2(v,\alpha)}{(\alpha,\alpha)}\alpha
$$
permutes $R$; and every <Cartan integer> $n_{\alpha\beta}=2(\alpha,\beta)/(\beta,\beta)$ is an <integer>. The restriction to a <reduced root system> and crystallographic integrality distinguishes roots of complex <semisimple Lie algebras> from more general reflection configurations.
For any two <roots of a root system>, the <Cauchy-Schwarz inequality> gives
$$
0\le n_{\alpha\beta}n_{\beta\alpha}=\frac{4(\alpha,\beta)^2}{(\alpha,\alpha)(\beta,\beta)}\le4.
$$
If the inner product is zero, both <Cartan integers> vanish. Otherwise their signs agree, and the absolute value of each is a positive <integer>. Dividing their product by an integer of absolute value at least one proves
$$
\boxed{0\le|n_{\alpha\beta}|\le4.}
$$
For nonproportional roots the product is strictly less than four. In a <reduced root system>, proportional roots are just $\pm\alpha$ and have Cartan integers $\pm2$, so the printed bound is intentionally looser than the resulting bound of three.
A <fundamental system of a root system> is a <basis> $\Delta$ of $E$ made of roots, such that each root is an integer combination of $\Delta$ with either all coefficients nonnegative or all nonpositive. Its members are the <simple roots>. Suppose distinct $\alpha,\beta\in\Delta$ had $n_{\alpha\beta}>0$. Then
$$
s_\beta(\alpha)=\alpha-n_{\alpha\beta}\beta\in R
$$
has a positive coefficient of $\alpha$ and a negative coefficient of $\beta$, contradicting the defining sign condition. Thus $n_{\alpha\beta}\le0$. Distinct <simple roots> are linearly independent, so their Cartan-integer product is strictly less than four. Combining integrality and the sign condition gives
$$
\boxed{n_{\alpha\beta}\in\{0,-1,-2,-3\}\quad(\alpha\ne\beta\text{ simple}).}
$$
Here \b[nonpositive] is the intended sense of the printed convention that includes zero among “negative” numbers; orthogonal simple roots really do give zero.
To form a <Dynkin diagram>, place a vertex at each <simple root>. Join two vertices by $n_{\alpha\beta}n_{\beta\alpha}$ bonds, hence zero, one, two or three. A multiple bond has an arrow toward the <short root>. Indeed $n_{\alpha\beta}/n_{\beta\alpha}=(\alpha,\alpha)/(\beta,\beta)$ determines the squared length ratio, and the diagram with the <Cartan matrix> reconstructs the angles and relative lengths. A single bond joins equal-length roots.
The connected finite <Dynkin diagrams> are the following. The descriptions include bond multiplicities and arrow directions, so distinguish dual diagrams:
* <An Dynkin diagram>, $A_n$ for $n\ge1$: a chain of $n$ vertices with only single bonds.
* <Bn Dynkin diagram and affine extension>, $B_n$ for $n\ge2$: a chain whose last bond is double, with its arrow toward the terminal short root; all earlier bonds are single. Only its finite diagram is used here.
* <Cn Dynkin diagram>, $C_n$ for $n\ge3$: the same chain with the double-bond arrow toward the penultimate short root and away from the terminal long root. $C_2$ and $B_2$ describe the same rank-two type after relabelling.
* <Dn Dynkin diagram>, $D_n$ for $n\ge4$: a simply laced tree with one trivalent vertex and arms of lengths $1,1,n-3$, counting edges.
* <En Dynkin diagram>, $E_6,E_7,E_8$: simply laced trees with a trivalent vertex and arms of lengths respectively $(1,2,2)$, $(1,2,3)$, $(1,2,4)$.
* <F4 Dynkin diagram>, $F_4$: a chain of four vertices, with a double central bond and two single outer bonds. Two consecutive vertices are long and two are short; the arrow goes from the long pair toward the short pair.
* <G2 Dynkin diagram>, $G_2$: two vertices joined by a triple bond, with arrow toward the short root.
There are no other connected finite <Dynkin diagrams>. Low-rank conventions also identify $B_1=C_1=A_1$ and $D_3=A_3$; $D_2$ is disconnected, so introduces no further connected type. Affine diagrams are outside this finite classification.
Finally suppose the underlying graph contained a cycle on $m\ge3$ distinct <simple roots> $\alpha_1,\ldots,\alpha_m$. Put $u_i=\alpha_i/\|\alpha_i\|$. Any bonded pair has
$$
(u_i,u_j)=-\frac12\sqrt{n_{\alpha_i\alpha_j}n_{\alpha_j\alpha_i}}\le-\frac12,
$$
and all other distinct pairs have nonpositive inner products. The cycle contributes at least $m$ bonded pairs, so
$$
\left\|\sum_{i=1}^m u_i\right\|^2=m+2\sum_{i<j}(u_i,u_j)\le m-m=0.
$$
But the <simple roots>, and hence these normalized vectors, are linearly independent, making the displayed sum nonzero. Positive definiteness gives a contradiction. Thus \b[the underlying graph of a finite Dynkin diagram has no cycle]. This <acyclicity of a finite Dynkin diagram> argument also excludes cycles with extra chords or multiple bonds; multiple bonds themselves are not treated as two-edge cycles.
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