= Solution
The <anti-Hermitian> operators represent the compact real form; its complexification also contains the Hermitian $H_i$ and the <raising operators> and <lowering operators> $E_\pm^r$, with $(E_+^r)^\dagger=E_-^r$. Define the <roots> in the two-dimensional Cartan plane by
$$
\alpha_1=(1,0),\qquad
\alpha_2=(-1/2,\sqrt3/2),\qquad
\alpha_3=\alpha_1+\alpha_2=(1/2,\sqrt3/2).
$$
Each has squared length one. For each $r=1,2,3$, set
$$
J_3^{(r)}=\alpha_r\cdot(H_1,H_2),\qquad
J_\pm^{(r)}=\sqrt2E_\pm^r.
$$
The displayed <commutators> give $[J_3^{(r)},J_\pm^{(r)}]=\pm J_\pm^{(r)}$ and $[J_+^{(r)},J_-^{(r)}]=2J_3^{(r)}$. Thus these are the three <root SU(2) subalgebras of SU(3)>.
The commuting Hermitian operators $H_1,H_2$ can be simultaneously diagonalized. A <weight of a representation> is a pair $\lambda=(h_1,h_2)$ for which
$$
H_i|\lambda,a\rangle=h_i|\lambda,a\rangle,
$$
where $a$ labels independent states of that <representation weight>. The <weight multiplicity> is the <dimension> of this simultaneous <eigenspace>. Commuting $H_i$ past a <ladder operator> proves
$$
\boxed{E_\pm^r:V_\lambda\longrightarrow V_{\lambda\pm\alpha_r},}
$$
with zero image allowed when the raised or lowered <representation weight> is absent.
Choose $\alpha_1,\alpha_2$ as positive <simple roots>. A <highest-weight vector> is a nonzero simultaneous eigenvector annihilated by $E_+^1,E_+^2,E_+^3$; the third condition follows from the first two and their <commutator>. In an <irreducible representation>, its <highest weight> $\Lambda$ has multiplicity one and repeated lowering generates the representation. The three <root SU(2) subalgebras of SU(3)> give nonnegative integer <Dynkin labels>
$$
p=2\Lambda\cdot\alpha_1=2h_1,\qquad
q=2\Lambda\cdot\alpha_2=-h_1+\sqrt3h_2.
$$
Equivalently, with $\omega_1=(1/2,1/(2\sqrt3))$ and $\omega_2=(0,1/\sqrt3)$,
$$
\boxed{\Lambda=p\omega_1+q\omega_2
=\left(\frac p2,\frac{p+2q}{2\sqrt3}\right),\qquad
p,q\in\mathbb Z_{\geq0}.}
$$
These are the <SU(3) highest-weight coordinates> in the normalization used here.
The <Weyl group> is generated by the reflections $s_r(\lambda)=\lambda-2(\lambda\cdot\alpha_r)\alpha_r$. The <SU(2) Lie algebra> root-string symmetry makes these reflections preserve both <representation weights> and their <weight multiplicities>. They generate the symmetries of an equilateral triangle: rotations through $120^\circ$ and three reflections. The outer polygon is the convex hull of the <Weyl group> orbit of $\Lambda$. When $p,q>0$ it is a hexagon with alternating side lengths $p,q$ in root-step units; when exactly one is zero it is a triangle; when both vanish it is a point. The representation conjugate to $(p,q)$ has labels $(q,p)$ and negative <representation weights>. For $p=q$, this gives an additional inversion symmetry and a regular hexagon with $60^\circ$ rotations. A general unequal-sided hexagon need not have that extra symmetry.
The <shell multiplicities in an SU(3) weight diagram> can be stated precisely. Let $P_{a,b}$ be the convex hull of the <Weyl group> orbit of $a\omega_1+b\omega_2$. For $\lambda$ in the <root lattice> coset $\Lambda+Q$,
$$
\boxed{m_{p,q}(\lambda)
=\sum_{k=0}^{\min(p,q)}
\mathbf1_{P_{p-k,q-k}}(\lambda).}
$$
<representation weights> outside this coset or outside the outer polygon are absent. This states that boundary <representation weights> have multiplicity one, each inward shell adds one, and the innermost triangular region has constant multiplicity $\min(p,q)+1$. When $p=q$ the innermost region is a single point. All <representation weights> of a triangular diagram have multiplicity one. Counting the diagram gives $\dim(p,q)=(p+1)(q+1)(p+q+2)/2$. The examples below display both triangular orientations, a regular hexagon, an unequal-sided hexagon, a central multiplicity three, and the trivial diagram.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-50-su3-weight-diagrams.png]
{title=SU(3) weight diagrams, including quark and adjoint weights and shell multiplicities}
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