= Cross-product Lie algebra
{title2=$(\mathbb R^3,\times)$}
The <vector space> $\mathbb R^3$ with the <cross product> as <Lie bracket>, $[x,y]=x\times y$. Bilinearity and $x\times x=0$ give an <Alternating Lie bracket>; the <vector triple product identity> proves the <Jacobi identity> because the three expressions $x\times(y\times z)=y(x\cdot z)-z(x\cdot y)$ cancel cyclically, using the symmetric <dot product>.
Its relation to the <SO(3) Lie algebra> is explicit. For a <vector> $x=(x_1,x_2,x_3)$, define the <skew-symmetric matrix>
$$
\widehat{x}=\begin{pmatrix}0&-x_3&x_2\\x_3&0&-x_1\\-x_2&x_1&0\end{pmatrix},\qquad \widehat{x}v=x\times v.
$$
Then $[\widehat{x},\widehat{y}]=\widehat{x\times y}$: acting on $v$, the left side is $x\times(y\times v)-y\times(x\times v)=(x\times y)\times v$. Every real $3\times3$ <skew-symmetric matrix> occurs uniquely, so this hat map is a <Lie algebra isomorphism>. With the standard <basis vectors> $e_1,e_2,e_3$,
$$
X=\widehat{e_1}=\begin{pmatrix}0&0&0\\0&0&-1\\0&1&0\end{pmatrix},\quad Y=\widehat{e_2}=\begin{pmatrix}0&0&1\\0&0&0\\-1&0&0\end{pmatrix},\quad Z=\widehat{e_3}=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix},
$$
and their <matrix commutators> are $[X,Y]=Z$, $[Y,Z]=X$, $[Z,X]=Y$. With active <rotation matrices> acting on column <vectors>, $\exp(t\widehat{x})$ rotates through angle $t\lVert x\rVert$ about the oriented axis $x$ when $x\ne0$. In particular $\exp(tZ)e_1=(\cos t,\sin t,0)$, fixing the positive-angle convention used in the <small rotation commutator>.
The map $x\mapsto-(i/2)x\cdot\sigma$ is also a <Lie algebra isomorphism> to the compact real <SU(2) Lie algebra>, using the <Pauli matrix commutator identity>. These two identifications concern <Lie algebras>; they do not identify the corresponding <Lie groups> globally.
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