I cannot understand the mixed product $$ (A\otimes B) (C\otimes D) = (AC) \otimes (BD) $$
I try to follow from the bases:
$$ \begin{eqnarray} \left(A\otimes B \right) \left(C \otimes D \right) &=& A_{ij}B_{kl}C_{mn}D_{pr}\ (e_i\otimes e_j \otimes e_k \otimes e_l) \ (e_m \otimes e_n \otimes e_p \otimes e_r) \\ &=& A_{ij}B_{kl}C_{mn}D_{pr}\ \delta_{lm} (e_i\otimes e_j \otimes e_k \otimes e_n \otimes e_p\otimes e_r) \\ &=& A_{ij}B_{kl}C_{ln}D_{pr}\ (e_i\otimes e_j \otimes e_k \otimes e_n \otimes e_p\otimes e_r) \end{eqnarray} $$ which is not equal to the correct one: $$ \begin{eqnarray} (AC) \otimes (BD) &=& (A_{ij} C_{mn}\ e_i \otimes e_j\ e_m \otimes e_n) \otimes (B_{kl} D_{pr}\ e_k \otimes e_l\ e_p \otimes e_r) \\ &=& (A_{ij} C_{mn} \delta_{jm}\ e_i \otimes e_n) \otimes (B_{kl} D_{pr} \delta_{lp} \ e_k \otimes e_r) \\ &=& (A_{ij} C_{jn} \ e_i \otimes e_n) \otimes (B_{kl} D_{lr} \ e_k \otimes e_r) \\ &=& A_{ij} C_{jn} B_{kl} D_{lr} \ e_i \otimes e_n \otimes \ e_k \otimes e_r \end{eqnarray} $$
Could you please help?