Cross-product Lie algebra ($({\mathbb{R}}^3,\times )$)

The vector space ${\mathbb{R}}^3$ with the cross product as Lie bracket. The vector triple-product identity proves the Jacobi identity. The map $x\mapsto -(i/2)x\cdot \sigma$ is a Lie algebra isomorphism to the compact real SU(2) Lie algebra, using the Pauli matrix commutator identity.