= Isotropic rank-three tensor and reflections
{title2=$T_{ijk}=a\epsilon_{ijk}\quad\text{under }SO(3)$}
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: $R_{ia}R_{jb}R_{kc}\epsilon_{abc}=(\det R)\epsilon_{ijk}$. 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.
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