Rotation generator ($J_k$)

In the three-dimensional vector representation, the Hermitian matrices $(J_k{)}_{ij}=-i{ϵ}_{kij}$ generate active rotations by the matrix exponential $R(\theta ,\mathbf{n})=\exp (-i\theta n^k{J}_k)$. They satisfy $[J_i,J_j]=i{ϵ}_{ijk}{J}_k$. The real antisymmetric matrices $A_k=-i{J}_k$ are the corresponding Lie algebra generators of $SO(3)$. Their commutator gives the leading change produced by interchanging two infinitesimal rotations.