Isotropic rank-three tensor and reflections ($T_{ijk}=a{ϵ}_{ijk}\text{under }SO(3)$)

In three-dimensional Euclidean space, a rank-three tensor invariant under every proper rotation is a scalar multiple of the Levi-Civita symbol. The distinction between proper rotations and all orthogonal transformations matters: $R_{ia}{R}_{jb}{R}_{kc}{ϵ}_{abc}=(\det R){ϵ}_{ijk}$. A polar tensor of odd rank invariant under every orthogonal transformation must vanish, since the transformation $R=-I$ changes its sign. A nonzero invariant rank-three object under reflections is instead a pseudotensor, whose transformation law includes the extra determinant factor.