= Axisymmetric second-order tensor
{title2=$T_{ij}=\alpha(\delta_{ij}-n_in_j)+\beta n_in_j+\omega\epsilon_{ijk}n_k$}
A Cartesian <tensor> invariant under all proper <rotations> fixing a unit <vector> $n$ is axisymmetric. Without assuming a <symmetric tensor>, its general second-order form is the displayed combination of the transverse identity, axial projector and axial antisymmetric part. For $n=e_3$ this is
$$
T=\begin{pmatrix}\alpha&\omega&0\\-\omega&\alpha&0\\0&0&\beta\end{pmatrix}.
$$
Indeed, imposing $RTR^T=T$ for a quarter-turn about $e_3$ forces $T_{11}=T_{22}$, $T_{12}=-T_{21}$ and all mixed axial-transverse entries to vanish. The planar block is a linear combination of the identity and the planar rotation generator, so it commutes with every rotation about the axis. Thus quarter-turn invariance already suffices for a second-order tensor. A <symmetric tensor> has $\omega=0$, recovering the <axisymmetric symmetric rank-two tensor> form. The antisymmetric term is allowed because only proper rotations about the fixed axis are imposed.
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