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    $\begingroup$ First of all, these are isotropic, not isomorphic tensors. You may find the result in this paper Harold Jeffreys (1973) On isotropic tensors relevant. It turns out they can be expressed as products of $\delta_{ij}$ and $\epsilon_{ijk}$. Finally, you are searching a fully-symmetric representation of $SO(3)$. Thus, group-theoretical methods apply. A computer algorithm is described in this paper $\endgroup$ Commented Feb 19, 2021 at 18:58
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    $\begingroup$ The general algorithm was developed by Coope et al. (1965, 1970). (aip.scitation.org/doi/10.1063/1.1697123, aip.scitation.org/doi/10.1063/1.1665190) However, to my knowledge, no public code which implement this algorithm is available. $\endgroup$ Commented Feb 19, 2021 at 19:12