I seek for the following relationship (if there is one so):
$$C \otimes D = (A_1 \otimes B_1) + (A_2 \otimes B_2)$$
I would like to obtain $C = f(A_1,A_2)$ (in terms of $A$'s) and $D = g(B_1,B_2)$ (in terms of $B$'s). For simplicity, we can assume $A_i$ and $B_i$ are covariance matrices, so positive-definite, square, and symmetric.
Any help is greatly appreciated!
PS: For more simplification (if so), we can assume $\dim(A_1) = \dim(A_2)$ and same for $B_i$'s.