= Solution
A reduced crystallographic <root system> $\Phi$ is a finite spanning subset of a real Euclidean <vector space>, excluding zero, such that $\Phi\cap\mathbb R\alpha=\{\pm\alpha\}$, each <reflection> $s_\alpha(v)=v-2(v,\alpha)\alpha/(\alpha,\alpha)$ preserves $\Phi$, and all Cartan integers $\langle\beta,\alpha\rangle=2(\beta,\alpha)/(\alpha,\alpha)$ are integers. A base is a vector-space <basis> $\Delta\subset\Phi$ in which each <root> has integer coordinates either all nonnegative or all nonpositive. The <Weyl group> is generated by the <root> reflections.
For nonparallel roots put $p=\langle\beta,\alpha\rangle$ and $q=\langle\alpha,\beta\rangle$. Then $pq=4\cos^2\theta$ is an integer strictly below $4$ and at least zero. Therefore
$$
\boxed{\cos\theta\in\{0,\pm\tfrac12,\pm\tfrac1{\sqrt2},\pm\tfrac{\sqrt3}{2}\}.}
$$
If $(\alpha,\beta)>0$, the positive integers $p,q$ have product $1,2$ or $3$, so at least one is one. If $p=1$, <reflection> gives $\beta-\alpha\in\Phi$ and hence its negative. If $q=1$, <reflection> directly gives $\alpha-\beta\in\Phi$. This proves the positive-inner-product assertion.
To prove that an $\alpha$-string has no gaps, suppose consecutive occurring integer positions $a<b$ in $\beta+t\alpha$ had $b-a\ge2$. If $(\beta+a\alpha,\alpha)<0$, apply the previous result to $\beta+a\alpha$ and $-\alpha$ to obtain the missing <root> at $a+1$. Thus this <inner product> must be nonnegative. Similarly the absence of a <root> at $b-1$ forces $(\beta+b\alpha,\alpha)\le0$. But the latter <inner product> exceeds the former by $(b-a)(\alpha,\alpha)>0$, impossible. Thus every intermediate integer position occurs.
Write the complete finite string as $\beta-r\alpha,\ldots,\beta+s\alpha$. <Reflection> maps position $t$ to $-p-t$, so it interchanges the two endpoints. Hence
$$
\boxed{r-s=\langle\beta,\alpha\rangle.}
$$
Here $r,s$ are the maximal endpoint distances, not arbitrary shorter distances mentioned in the existence assertion.
Use the Cartan-<matrix> convention $C_{ij}=\langle\alpha_i,\alpha_j\rangle$. The displayed <matrix> is realized by
$$
\alpha_1=e_2,\qquad\alpha_2=e_1-e_2,
$$
with the standard orthonormal coordinates. Closure under the two simple reflections gives
$$
\boxed{\Phi=\{\pm e_1,\pm e_2,\pm(e_1-e_2),\pm(e_1+e_2)\}.}
$$
The positive roots are $\alpha_1,\alpha_2,\alpha_1+\alpha_2,2\alpha_1+\alpha_2$. Conversely the standard fact that every <root> is Weyl-conjugate to a <simple root>, or successive height reduction by simple reflections, shows these two <reflection> orbits exhaust the roots. This is the rank-two system $B_2$, also called $C_2$ after exchanging the length convention. With a transposed Cartan-<matrix> convention, interchange which simple-<root> label is short; the geometric system and <Weyl group> are unchanged.
\Image[/past-exam-of-the-lie-algebras-and-representation-theory-of-symmetric-groups-course-of-the-university-of-oxford/2010-c21-b2-roots.png]
{title=Rank-two root system with its chosen simple roots}
The <reflection> in $e_2$ flips the second coordinate, while the <reflection> in $e_1-e_2$ swaps the two coordinates. These generate all eight signed permutation matrices in two coordinates. Thus
$$
\boxed{W(\Phi)\cong D_8,\quad |W(\Phi)|=8,}
$$
where $D_8$ means the symmetry group of a square.
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