= Solution
The <Weyl dimension formula> is increasing in each nonnegative <Dynkin label>. If $a\geq3$ or $b\geq3$, the dimension is at least $d(3,0)=10$. Among $0\leq a,b\leq2$, the cases with both labels nonzero, other than $(1,1)$, have dimension at least $d(2,1)=15$. Thus the following is the complete list of <irreducible representations> of dimension strictly less than $10$:
$$
\boxed{\begin{array}{c|c|r|r|r}
(a,b)&R&d&C&c\\\hline
(0,0)&\mathbf1&1&0&0\\
(1,0)&\mathbf3&3&8/3&1\\
(0,1)&\overline{\mathbf3}&3&8/3&1\\
(2,0)&\mathbf6&6&20/3&5\\
(0,2)&\overline{\mathbf6}&6&20/3&5\\
(1,1)&\mathbf8&8&6&6
\end{array}}
$$
Here $\mathbf6=\operatorname{Sym}^2\mathbf3$ is the <two-index symmetric representation> and $\mathbf8$ is the <Adjoint representation>. Interchanging $a$ and $b$ gives the <conjugate representation>, explaining equal dimensions, <Casimir eigenvalues> and <Dynkin indices> in each conjugate pair. In the alternative convention of <2/b/solution>, divide the $C$ and $c$ columns by two.
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