Cross-product model of su(2) ($\mathfrak{su}(2)\cong ({\mathbb{R}}^3,\times )$)

The real vector space of traceless skew-Hermitian matrices of size two is the Lie algebra $\mathfrak{su}(2)$. For the basis $M_1=\frac{1}{2}\mathrm{diag}(i,-i)$, $M_2=\frac{1}{2}\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, $M_3=\frac{1}{2}\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right)$, cyclic commutators satisfy $[M_1,M_2]=M_3$. The map $a\mapsto \sum a_j{M}_j$ carries the cross product to the commutator. Every target is a single commutator: choose a unit vector perpendicular to its associated vector $c$ and use $u\times (c\times u)=c$.