= Solution
Work with a finite-dimensional complex <semisimple Lie algebra>. The <Dynkin diagram> is obtained from its intrinsic <root system>, with several structure theorems entering the construction.
First choose a <Cartan subalgebra> $\mathfrak h$. The semisimple structure theory gives the <root-space decomposition>
$$
\mathfrak g=\mathfrak h\oplus\bigoplus_{\alpha\in\Phi}\mathfrak g_\alpha,\qquad
\mathfrak g_\alpha=\{X:[H,X]=\alpha(H)X\text{ for all }H\in\mathfrak h\}.
$$
The nonzero <roots> span $\mathfrak h^*$, <root spaces> have <dimension> one, and every <root> has its negative. The rank is $r=\dim\mathfrak h$.
Next use the <Killing form>. Its nondegeneracy on $\mathfrak h$ and its positive-definite restriction to the real <coroot> span identify the real <root> span $E=\mathbb R\Phi$ with a Euclidean space. Explicitly, if $t_\lambda=\mathcal B^{-1}(\lambda)$, put $(\lambda,\mu)=B(t_\lambda,t_\mu)$. The <sl2 subalgebra associated with a root> supplies its normalized <coroot> $H_\alpha$ and the integral pairing
$$
\beta(H_\alpha)=\frac{2(\beta,\alpha)}{(\alpha,\alpha)}\in\mathbb Z.
$$
The <root-string theorem> makes the reflection $s_\alpha(\beta)=\beta-\beta(H_\alpha)\alpha$ permute $\Phi$. Together with the absence of multiples other than $\pm\alpha$, this shows that $\Phi$ is a finite <reduced root system> and a <crystallographic root system>. These properties come from <semisimple Lie algebra> theory, not from the diagram's definition.
Choose a vector in $E$ that is not perpendicular to any <root>. Define the <positive roots> by positive inner product with this vector. The <simple roots> are those <positive roots> that cannot be expressed as sums of two <positive roots>. Root-system theory shows that they form a <basis> $\Delta=\{\alpha_1,\ldots,\alpha_r\}$, and each <root> is an integer combination of them with coefficients all of one sign. Distinct <simple roots> have nonpositive inner product.
Use the standard <Cartan matrix> convention
$$
A_{ij}=\alpha_i(H_{\alpha_j})=\frac{2(\alpha_i,\alpha_j)}{(\alpha_j,\alpha_j)}.
$$
There is one diagram vertex for each <simple root>. For $i\ne j$, the integral off-diagonal entries are nonpositive and
$$
A_{ij}A_{ji}=4\cos^2\theta_{ij}\in\{0,1,2,3\}.
$$
Connect the two vertices with respectively zero, one, two or three edges. The corresponding angles are $90^\circ,120^\circ,135^\circ,150^\circ$. A multiple edge carries an arrow \b[toward the shorter <root>]. For joined vertices, the length ratio is determined by $A_{ij}/A_{ji}=(\alpha_i,\alpha_i)/(\alpha_j,\alpha_j)$; hence the diagram records both angle and relative length.
For example, $\mathfrak{sl}_3$ has <simple roots> $L_1-L_2$ and $L_2-L_3$, giving two equal-length vertices joined by one edge: type $A_2$. For $\mathfrak{sp}_4$, take $\alpha_1=L_1-L_2$ and $\alpha_2=2L_2$. Then
$$
A=\begin{pmatrix}2&-1\\-2&2\end{pmatrix},
$$
so the two vertices have a double edge directed toward $\alpha_1$, the shorter <root>: type $C_2$.
Finally, orthogonal irreducible components of the <root system> correspond to the simple <ideals of a Lie algebra> in the <semisimple Lie algebra>, so its diagram is the disjoint union of their connected diagrams. Conjugacy of <Cartan subalgebras> and the <Weyl group> action on choices of <positive roots> make the resulting diagram independent of these choices up to isomorphism. \b[Thus the vertices, edge multiplicities, arrows and connected components encode the simple-root geometry and the simple-ideal decomposition.]
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