= Solution
Every traceless <skew-Hermitian matrix> of size two has the form
$$
C=\begin{pmatrix}ip&w\\-\overline w&-ip\end{pmatrix},\qquad p\in\mathbb R,
$$
so it equals $\mathcal M(c)$ for $c=(2p,2\operatorname{Re}w,2\operatorname{Im}w)$. If $c=0$, take $A=B=0$. Otherwise choose a <unit vector> $u$ perpendicular to $c$ and set $a=u$, $b=c\times u$. The <vector triple product> gives $a\times b=c$.
By part (iii), choosing $A=\mathcal M(a)$ and $B=\mathcal M(b)$ gives \b[the required representation]
$$
\boxed{C=[A,B].}
$$
In fact both factors can themselves be chosen as traceless <skew-Hermitian matrices>. The <cross-product model of su(2)> explains this geometric <commutator> construction.
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