= Solution
The triple bond in the original <Dynkin diagram> gives the <Cartan integers>
$$
\langle\beta,\alpha^\vee\rangle=-1,\qquad\langle\alpha,\beta^\vee\rangle=-3,
$$
where $\alpha$ is the <long root> and $\beta$ the <short root>. Their product is $4\cos^2\theta=3$, and the <inner product> of distinct <simple roots> is nonpositive. The ratio of the two integers gives the squared-length ratio. Therefore
$$
\boxed{\theta=\frac{5\pi}{6}=150^\circ,\qquad\frac{\|\alpha\|}{\|\beta\|}=\sqrt3.}
$$
It is convenient to normalize $(\alpha,\alpha)=6$, $(\beta,\beta)=2$ and $(\alpha,\beta)=-3$. Scaling the <inner product> does not affect the <root system> or the fundamental-weight relations.
For nonproportional roots $\gamma,\delta$, the <root-string theorem> states that the <root string> is consecutive:
$$
S_{\gamma,\delta}=\{\delta-p\gamma,\ldots,\delta+q\gamma\},\qquad p,q\in\mathbb Z_{\ge0},\qquad p-q=\frac{2(\delta,\gamma)}{(\gamma,\gamma)}.
$$
The endpoints are maximal, and reflection in $\gamma$ reverses the string. The <Cartan integer> determines $p-q$, not in general the total $p+q+1$ by itself. This result follows by restricting the <Adjoint representation> to the <sl2 subalgebra associated with a root>. If the roots are distinct <simple roots>, $\delta-\gamma$ cannot be a root: its simple-root coefficients have opposite signs. Thus $p=0$ and
$$
\boxed{|S_{\gamma,\delta}|=1-\langle\delta,\gamma^\vee\rangle.}
$$
Here length means number of roots; the number of intervals between them is one less. Distinctness matters. If $\gamma=\delta$, a reduced <root system> gives the set $\{\gamma,-\gamma\}$, with a missing zero between them, so the consecutive-string theorem and the displayed simple-root formula do not apply.
In the <G2 root system>, the initial strings are
$$
\boxed{S_{\alpha,\beta}=\{\beta,\alpha+\beta\},\qquad S_{\beta,\alpha}=\{\alpha,\alpha+\beta,\alpha+2\beta,\alpha+3\beta\}.}
$$
For $\delta=\alpha+3\beta$, $\langle\delta,\alpha^\vee\rangle=-1$. Since $\delta-\alpha=3\beta$ is not a root in a <reduced root system>, $p=0$ and $q=1$: this generates $2\alpha+3\beta$. The remaining strings explain why the construction stops. The $\alpha$-strings through $\beta$ and $\alpha+\beta$ are the same two-element string; the one through $\alpha+2\beta$ is a singleton since subtracting $\alpha$ gives $2\beta$, and its <Cartan integer> is zero; the one through $2\alpha+3\beta$ is the string $\{\alpha+3\beta,2\alpha+3\beta\}$. The $\beta$-strings through $\alpha$, $\alpha+\beta$, $\alpha+2\beta$ and $\alpha+3\beta$ are the initial four-element string. Finally, $2\alpha+3\beta$ is orthogonal to $\beta$; its $\beta$-string is a singleton because $2\alpha+2\beta=2(\alpha+\beta)$ is not a root. Apply the same reasoning to negatives. Using the permitted completeness of this procedure gives
$$
\boxed{\Phi=\pm\{\beta,\alpha,\alpha+\beta,\alpha+2\beta,\alpha+3\beta,2\alpha+3\beta\}.}
$$
The short positive roots are $\beta,\alpha+\beta,\alpha+2\beta$, of squared length two; the other three are long, of squared length six. Each <root space> is one-dimensional and the <Cartan subalgebra> has dimension two, so
$$
\boxed{\dim G_2=2+12=14.}
$$
Here the dimension refers to the <Lie algebra>, with one Cartan generator per rank, not just the number of roots.
Write a prospective weight as $w=a\alpha+b\beta$. The pairings with simple <coroots> are
$$
\langle w,\alpha^\vee\rangle=2a-b,\qquad\langle w,\beta^\vee\rangle=-3a+2b.
$$
The <fundamental weights> are dual to those <coroots>. Solving the two linear systems gives, in the long-root-first numbering of this paper,
$$
\boxed{\omega_1=2\alpha+3\beta,\qquad\omega_2=\alpha+2\beta.}
$$
The representation with <Dynkin labels> $(0,1)$ has <highest weight> $\lambda=\omega_2$, a short root. Numbering the short root first, as some references do, would call this the $(1,0)$ representation instead; the representation itself is unchanged.
The weight set of a finite-dimensional irreducible <highest-weight representation> is invariant under the <Weyl group> and lies in the <convex hull> of the orbit of its <highest weight>. All weights also differ from the <highest weight> by an element of the <root lattice>. The orbit of $\lambda$ comprises the six short roots, so all six are weights. The <lowering operators> give the chain
$$
\alpha+2\beta\ \xrightarrow{-\beta}\ \alpha+\beta\ \xrightarrow{-\alpha}\ \beta\ \xrightarrow{-\beta}\ 0\ \xrightarrow{-\beta}\ -\beta\ \xrightarrow{-\alpha}\ -\alpha-\beta\ \xrightarrow{-\beta}\ -\alpha-2\beta.
$$
In particular zero occurs: at weight $\beta$, its pairing with $\beta^\vee$ is two, so the lowering operator is nonzero by the finite-dimensional <sl2 Lie algebra> representation theory. The $\beta$-string through it is the usual three-weight string $\beta,0,-\beta$; higher weight $2\beta$ would lie outside the highest-weight convex hull.
There can be no further weights. Every point of that convex hull has squared norm at most $\|\lambda\|^2=2$, while a root-lattice point has
$$
\|a\alpha+b\beta\|^2=6a^2-6ab+2b^2=\frac32a^2+2\left(b-\frac32a\right)^2.
$$
The integer solutions of the bound $\|w\|^2\le2$ are precisely zero and the six short roots: $a=0$ gives $b=-1,0,1$; $a=1$ gives $b=1,2$; and $a=-1$ gives $b=-1,-2$. Thus
$$
\boxed{\operatorname{Wt}V(0,1)=\{0,\pm\beta,\pm(\alpha+\beta),\pm(\alpha+2\beta)\},\qquad\dim V(0,1)=7.}
$$
The final dimension uses the stipulated nondegeneracy of the weights. It also agrees with the <Weyl dimension formula> without that stipulation. The zero weight is not in the Weyl orbit of the nonzero weights, so this is not a <minuscule representation>.
\Image[/media/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-302-g2-roots.png]
{title=G2 roots and weights of its seven-dimensional representation}
{description=The twelve <roots of a root system> and seven <weights> in the long-root-first convention.}
{height=500}
In the diagram above, a coordinate label $(a,b)$ means $a\alpha+b\beta$. The left panel contains all twelve <roots of a root system>; the right panel contains the six short-root weights and the zero weight. The long-root-first convention is the same as in the calculations.
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