Oriented-axis invariant Cartesian tensor ($T_{ij}=\alpha ({\delta }_{ij}-n_i{n}_j)+\beta n_i{n}_j+\omega {ϵ}_{ijk}{n}_k$)

A Cartesian second-rank tensor invariant under all rotations about an oriented unit axis $n$ has the displayed form. A half-turn about the axis eliminates the mixed axial-transverse entries, and a quarter-turn makes the transverse block a scalar matrix plus an antisymmetric planar rotation generator. Conversely both transverse terms commute with every rotation about the axis. The antisymmetric term is allowed unless symmetry or an additional axis-reversing rotation is imposed. This differs from an axially invariant second-rank tensor for an unoriented axis.