I have been working on a calculation that involves the gradient of the electric field, specifically using the tensor product of the electric field gradient tensor $\nabla E^* \otimes \nabla E$, and I am a little confused on some details about working with these objects. For completeness sake, in cartesian form $\nabla E = \begin{bmatrix}\partial_x E_x & \partial_x E_y & \partial_x E_z\\ \partial_y E_x & \partial_y E_y & \partial_y E_z\\ \partial_z E_x & \partial_z E_y & \partial_z E_z\end{bmatrix}$.
With regards to constructing the irreducible spherical tensor from a cartesian tensor, I should expect that there should be 3 resulting tensors, each with $2k+1$ components as it is a rank 2 tensor, so the 9 components: $\nabla E^{(0)}_0$,$\nabla E^{(1)}_{0,\pm1}$,$\nabla E^{(2)}_{0,\pm 1,\pm 2}$ which are constructed in the usual way for spherical tensors.
My question comes for computing the tensor product. I understand that I should end up with spherical tensors ranging from rank 0 to 4, and can compute each component using $$ T^{(k)}_q = \sum^{k_1}_{q_1=-k_1} \sum^{k_2}_{q_2=-k_2} A^{k_1}_{q_1}B^{k_2}_{q_2} \left<k_1 q_1;k_2 q_2|k_1 k_2; k,q\right> $$
and my confusion comes from how to know which $k_1$ and $k_2$ to use in the above expression as the electric field gradient tensor I have has components of 3 different ranks ($k_1$ = 0,1 or 2)?
In articles I see they only ever mention the $\nabla E^{(2)}$ components (for example page 5 of https://tf.nist.gov/general/pdf/1421.pdf), am I making a fundamental misunderstanding on how to work with these spherical tensors?
Apologies if the question is a little unclear, it is hard to know exactly what I am not understanding.