= Cross-product model of su(2)
{title2=$\mathfrak{su}(2)\cong(\mathbb R^3,\times)$}
= SU(2)-SO(3) Lie algebra isomorphism
{c}
{synonym}
The real vector space of traceless <skew-Hermitian matrices> of size two is the <Lie algebra> $\mathfrak{su}(2)$. For the basis $M_1=\frac12\operatorname{diag}(i,-i)$, $M_2=\frac12\begin{pmatrix}0&1\\-1&0\end{pmatrix}$, $M_3=\frac12\begin{pmatrix}0&i\\i&0\end{pmatrix}$, cyclic <commutators> satisfy $[M_1,M_2]=M_3$. The map $a\mapsto\sum a_jM_j$ 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 $u\times(c\times u)=c$.
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