= Solution
For a <Cartan-Weyl basis>, the <Jacobi identity> gives
$$
[H_i,[E_\alpha,E_\beta]]=[[H_i,E_\alpha],E_\beta]+[E_\alpha,[H_i,E_\beta]]
=(\alpha_i+\beta_i)[E_\alpha,E_\beta].
$$
Thus the bracket has joint Cartan weight $\alpha+\beta$. If $\beta=-\alpha$, it belongs to the zero-weight space, which is the <Cartan subalgebra>, and so is a <linear combination> $\widetilde\alpha_iH_i$. If $\alpha+\beta$ is a nonzero <root>, its <root space> is one-dimensional and the bracket is proportional to $E_{\alpha+\beta}$. If that nonzero sum is not a <root>, the bracket vanishes. This proves the asserted forms without assuming every pair of <roots> has a <root> sum.
With the stated normalization, put $H_\alpha=2\alpha_iH_i/\alpha^2$. The commutators become
$$
[H_\alpha,E_\alpha]=2E_\alpha,\qquad [H_\alpha,E_{-\alpha}]=-2E_{-\alpha},\qquad
[E_\alpha,E_{-\alpha}]=H_\alpha.
$$
The adjoint relations make $J_3=H_\alpha/2$, $J_+=E_\alpha$, $J_-=E_{-\alpha}$ the usual <SU(2) Lie algebra> generators. Hence every finite-dimensional <SU(2)> irreducible component has $J_3$ <eigenvalues> $m=-j,-j+1,\ldots,j$, with $2j$ an integer. Therefore \b[$H_\alpha$ has integer <eigenvalues> $2m$], including in the finite-dimensional <Adjoint representation>. Algebraically this is the <sl2 subalgebra associated with a root>.
If $[H_\alpha,X_\lambda]=\lambda X_\lambda$, another use of the <Jacobi identity> gives
$$
[H_\alpha,[E_\alpha,X_\lambda]]=(\lambda+2)[E_\alpha,X_\lambda].
$$
The bracket is therefore zero or lies in the <eigenvalue>-$\lambda+2$ space. Within one irreducible <SU(2)> ladder it is proportional to the next chosen vector $X_{\lambda+2}$. If the full eigenspace has multiplicity, the weight statement is the general one; proportionality to one named vector presupposes the ladder choice. Similarly $E_{-\alpha}$ lowers this <eigenvalue> by two, and the ladder weights run from $-n$ to $n$ in steps of two.
For a <root vector> $E_\beta$, its $H_\alpha$ weight is
$$
m_{\alpha\beta}=\frac{2\alpha\cdot\beta}{\alpha^2}.
$$
The adjoint <SU(2)> ladder therefore proves
$$
\boxed{\frac{2\alpha\cdot\beta}{\alpha^2}\in\mathbb Z.}
$$
For nonparallel $\alpha,\beta$, raising and lowering $E_\beta$ move its joint Cartan weight through the <root string> $\beta+k\alpha$. The finite irreducible ladder has weights symmetric about zero. Starting at $m_{\alpha\beta}$, reaching its partner $-m_{\alpha\beta}$ requires $k=-m_{\alpha\beta}$. The corresponding nonzero vector lies in the <root space> of $\beta-m_{\alpha\beta}\alpha$, proving the <Weyl reflection> property:
$$
\boxed{s_\alpha(\beta)=\beta-\frac{2\alpha\cdot\beta}{\alpha^2}\alpha\text{ is a root}.}
$$
There is no zero joint <root> in this string when $\alpha,\beta$ are nonparallel. Its one-dimensional <root spaces> prevent two irreducible ladders with the same weight parity from overlapping, so the ladder argument applies to the entire string. For $\beta=\pm\alpha$, reflection simply interchanges the two opposite <roots>, which already exist by the adjoint relation.
For angle and length restrictions, set $p=2\alpha\cdot\beta/\alpha^2$ and $q=2\alpha\cdot\beta/\beta^2$. These <Cartan integers> have the same sign when nonzero and obey
$$
pq=4\cos^2\theta,\qquad \frac{\alpha^2}{\beta^2}=\frac qp.
$$
For nonparallel, nonorthogonal <roots>, $pq$ is an integer strictly between zero and four. The possibilities are consequently:
* $pq=1$: $|p|=|q|=1$, equal lengths, and $\theta=\pi/3$ or $2\pi/3$.
* $pq=2$: $\{|p|,|q|\}=\{1,2\}$, squared-length ratio $2$ or $1/2$, and $\theta=\pi/4$ or $3\pi/4$.
* $pq=3$: $\{|p|,|q|\}=\{1,3\}$, squared-length ratio $3$ or $1/3$, and $\theta=\pi/6$ or $5\pi/6$.
<Orthogonal> <roots> have $p=q=0$ and $\theta=\pi/2$. Parallel <roots> have $pq=4$; the integer pairs would allow only length multiples $1,2,1/2$. The stated reducedness condition excludes the latter two, so parallel <roots> are equal or opposite, with angles zero or $\pi$ and equal lengths. Thus
$$
\boxed{\theta\in\{0,\pi/6,\pi/4,\pi/3,\pi/2,2\pi/3,3\pi/4,5\pi/6,\pi\}.}
$$
Not every <root system> realizes every listed angle.
The <orthogonal> pair alone gives no ratio through $q/p$. To finish the length restriction for a <simple Lie algebra>, use irreducibility of its <root system>. Let $U$ span the <Weyl group> orbit of $\alpha$. Both $U$ and $U^\perp$ are invariant under the <orthogonal> <root reflections>. If a <root> had nonzero projections in both, reflection of its projection in $U$ would have a nonzero $U^\perp$ component, violating invariance. Hence every <root> is entirely in one of these subspaces. A proper nonzero $U$ would split the <root system> into two <orthogonal> parts, contradicting irreducibility. This proves that the <Weyl orbit spans an irreducible root space>.
In particular, some orbit <root> $\gamma$ is nonorthogonal to any chosen $\beta$, and $\gamma^2=\alpha^2$. Applying the previous integer-pair restriction to $\gamma,\beta$ therefore also handles <orthogonal> $\alpha,\beta$:
$$
\boxed{\frac{\alpha^2}{\beta^2}\in\{1,2,1/2,3,1/3\}.}
$$
There can be at most two different <root> lengths: three lengths would have ratios two and three relative to the shortest, but then the longest-to-middle squared-length ratio would be $3/2$, which is forbidden. A reducible collection of <orthogonal> rank-one factors could have independently chosen scales; the simple-algebra assumption excludes that exception.
Choose a generic linear functional to define <positive roots>. The <simple roots> are those positive <roots> which cannot be written as sums of two positive <roots>; they form a basis in which all <root> coefficients are integers of one sign. Distinct simple <roots> have nonpositive inner product: if their inner product were positive, their <root> string would contain their difference, and whichever difference is positive would decompose one of them. In rank two, irreducibility excludes the <orthogonal> case. The allowed simple-<root> angles are therefore $120^\circ,135^\circ,150^\circ$, giving the three irreducible diagrams in the <classification of rank-two root systems>: <A2 root system>, <B2 root system> and <G2 root system>. Reflections in the two simple <roots> generate their respective six, eight and twelve <roots>.
Here are explicit simple-<root> choices and positive <roots>; adding their negatives gives each complete diagram.
* For <A2 root system>, take $\alpha_1=(1,0)$ and $\alpha_2=(-1/2,\sqrt3/2)$. The positive <roots> are $\alpha_1,\alpha_2,\alpha_1+\alpha_2$; all have squared length one.
* For <B2 root system>, take the long $\alpha_1=(1,-1)$ and short $\alpha_2=(0,1)$. The positive <roots> are $\alpha_1,\alpha_2,\alpha_1+\alpha_2,\alpha_1+2\alpha_2$. These give the four axis <roots> and four diagonal <roots>, with squared lengths one and two. The rank-two $C_2$ <Lie algebra> is isomorphic to this one; changing the long/short ordering changes its Cartan presentation.
* For <G2 root system>, take short $\alpha_1=(1,0)$ and long $\alpha_2=(-3/2,\sqrt3/2)$. The positive <roots> are $\alpha_1,\alpha_2,\alpha_1+\alpha_2,2\alpha_1+\alpha_2,3\alpha_1+\alpha_2,3\alpha_1+2\alpha_2$. They give two hexagons rotated by $30^\circ$, with squared lengths one and three.
The nonnegative coefficients in each positive-<root> list verify the chosen <simple roots> explicitly. The <orthogonal> four-<root system> $A_1\times A_1$ is rank two but reducible, so it is not a diagram for the simple algebra assumed here.
Define the <Cartan matrix> in the column-coroot convention:
$$
C_{ij}=\langle\alpha_i,\alpha_j^\vee\rangle=\frac{2\alpha_i\cdot\alpha_j}{\alpha_j^2},\qquad
[H_{\alpha_j},E_{\alpha_i}]=C_{ij}E_{\alpha_i}.
$$
Using the ordered <roots> above gives
$$
\boxed{C_{A_2}=\begin{pmatrix}2&-1\\-1&2\end{pmatrix},\qquad
C_{B_2}=\begin{pmatrix}2&-2\\-1&2\end{pmatrix},\qquad
C_{G_2}=\begin{pmatrix}2&-1\\-3&2\end{pmatrix}.}
$$
The alternative row-coroot convention transposes these <matrices>. Stating the convention and <root> ordering avoids a spurious disagreement in the unequal-length cases.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-63-rank-two-roots.png]
{title=The irreducible rank-two root systems A2, B2 and G2, with long and short roots and explicit ordered simple roots highlighted}
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