= Solution
Use the convention $A_{ij}=\langle\alpha_j,\alpha_i^\vee\rangle=2(\alpha_i,\alpha_j)/(\alpha_i,\alpha_i)$. For a finite reduced crystallographic <root system>, diagonal entries are two and off-diagonal entries belong to $\{0,-1,-2,-3\}$, with $A_{ij}A_{ji}\in\{0,1,2,3\}$. Transposing this convention merely transposes the <Cartan matrix>.
The $\alpha$-string through $\beta$ has the consecutive <roots of a root system>[roots] $\beta-p\alpha,\ldots,\beta+q\alpha$, where $p,q\ge0$ and $p-q=\langle\beta,\alpha^\vee\rangle$. For distinct <simple roots> $\alpha_i,\alpha_j$, the negative step is absent, so $p=0$ and $q=-A_{ij}$.
The <matrix> determines each simple <reflection> on the simple-<root of a root system>[root] <basis>:
$$
s_i(\alpha_j)=\alpha_j-A_{ij}\alpha_i.
$$
It therefore determines their generated <Weyl group> as linear transformations. Every <root of a root system>[root] is in the Weyl orbit of a <simple root>. For a positive nonsimple <root of a root system>[root] $\beta=\sum c_i\alpha_i$, some $(\beta,\alpha_i)$ is positive because $\sum c_i(\beta,\alpha_i)=(\beta,\beta)>0$. Its <reflection> subtracts a positive integer multiple of $\alpha_i$ and lowers height. It stays positive unless the <root of a root system>[root] is simple, since the other simple-<root of a root system>[root] coefficients are unchanged and at least one is positive. Induction on height reaches a <simple root>; negative <roots of a root system>[roots] follow by sign. Hence $\Phi=W\Delta$, which is completely reconstructed by those <reflection> <matrices>. This determines the system up to <root of a root system>[root]-system isomorphism, rather than fixing an absolute length scale; irreducible components may be rescaled.
For <positive roots for a chain Cartan matrix>, symmetry and adjacency imply all <simple roots> have equal length; normalize their squared lengths to two. Adjacent inner products are minus one and all others zero. Since every <root of a root system>[root] is Weyl-conjugate to a <simple root>, every <root of a root system>[root] also has squared length two. Write a <positive root> as $\beta=\sum_{i=1}^lc_i\omega_i$ with nonnegative integers $c_i$, and set $c_0=c_{l+1}=0$. Then
$$
2=(\beta,\beta)=2\sum_{i=1}^lc_i^2-2\sum_{i=1}^{l-1}c_ic_{i+1}=\sum_{i=0}^l(c_{i+1}-c_i)^2.
$$
There must be exactly two nonzero integer differences, one $+1$ and one $-1$. Nonnegativity and the two zero endpoint values imply that the coefficients are one on one consecutive interval and zero elsewhere.
Conversely a single <simple root> is a <root of a root system>[root]. If $\omega_i+\cdots+\omega_j$ is a <root of a root system>[root] and $j<l$, its pairing with $\omega_{j+1}^\vee$ is minus one. Reflecting in $\omega_{j+1}$ therefore adds that <simple root>, making the next longer interval a <root of a root system>[root]. Induction constructs every interval. Thus
$$
\boxed{\Phi^+=\left\{\sum_{k=i}^j\omega_k:1\le i\le j\le l\right\}.}
$$
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