I try to answer your question, although I'm not completely sure if I understand what you are asking for.
You want a decomposition like for example: $$ (pq|rs) = \sum_P A^p_P A^q_P A^r_P A^s_P $$ am I correct?
In the framework of density fitting this type of decomposition is not possible without further steps/tricks. If we look at the two electron integrals \begin{equation} (pq|rs)=\int d\pmb{r}1 \int d\pmb{r}_2 \phi_p(\pmb{r}_1) \phi_q(\pmb{r}_1) \frac{1}{\pmb{r}{12}} \phi_r(\pmb{r}2) \phi_s(\pmb{r}_2) \quad, \end{equation} we can consider this electron repulsion integral as repulsion between two generalized electron densities \begin{equation} (pq|rs)=\int d\pmb{r}_1 \int d\pmb{r}_2 \rho{pq}(\pmb{r}1) \frac{1}{\pmb{r}{12}} \rho_{rs}(\pmb{r}2) \quad, \end{equation} with $\rho{pq}(\pmb{r}) = \phi_p(\pmb{r}) \phi_q(\pmb{r})$ and $\rho_{rs}(\pmb{r}) = \phi_r(\pmb{r}) \phi_s(\pmb{r})$. This density maybe approximated with the aid of an auxiliary basis set: \begin{equation} \tilde{\rho}{pq}(\pmb{r}) = \sum^{N{aux}}P d_P^{pq} \chi_P(\pmb{r}) \end{equation} Hence the name "density fitting". The coefficients $d_P^{pq}$ are determined by minimizing the functional \begin{equation} \Delta{pq} = \int d\pmb{r}1 \int d\pmb{r}_2 \frac{\left[ {\rho}{pq}(\pmb{r_1}) - \tilde{\rho}{pq}(\pmb{r_1}) \right] \left[ {\rho}{rs}(\pmb{r_2}) - \tilde{\rho}{rs}(\pmb{r_2}) \right]}{r{12}} \end{equation} and leads according to <PERSON> [1] to: \begin{equation} d_Q^{pq} = \sum_{P} \left(pq|P\right) \left[\pmb{J}^{-1}\right]{PQ} \quad, \end{equation} with \begin{equation} \left(pq|P\right) = \int d\pmb{r}_1 \int d\pmb{r}_2 \phi_p(\pmb{r}_1) \phi_q(\pmb{r}_1) \frac{1}{\pmb{r}{12}} \chi_P(\pmb{r}2) \end{equation} \begin{equation} J{PQ} = \int d\pmb{r}1 \int d\pmb{r}_2 \chi_P(\pmb{r}_1) \frac{1}{\pmb{r}{12}} \chi_Q(\pmb{r}2) \end{equation} Then we can re-write the electron repulsion integral as \begin{equation} (pq|rs)=\sum{PQ} \left(pq|P\right) \left[\pmb{J}^{-1}\right]{PQ} \left(Q|rs\right) \end{equation} The framework of density fitting does not give us the possibility to decompose $\left(pq|P\right)$ further, but it is possible in combination with more general tensor decomposition techniques like the canonical decomposition (CP)[2], in which we approximate a tensor in the format you were fishing for (as long as I understood you correctly) \begin{equation} X{i,j,... 374