Cartesian isotropic tensors revisited

Abstract: An alternative, streamlined methodology is presented for deriving the isotropic Cartesian tensor bases under the special orthogonal group $\text{SO}(3)$. By shifting the traditional interpretation of the isotropy condition to analyze it as an algebraic system of equations for the rotation matrices themselves rather than the tensor components, lower-rank bases can be explicitly established in just a few lines without requiring finite coordinate rotations or complicated contractions of infinitesimal generators. Furthermore, a transparent combinatorial interpretation is provided for the number of tensors in the general higher-rank spanning sets. Finally, the Gram-matrix method is advocated as an efficient computational sieve to resolve the subsequent linear dependence.