Q:

Can this block matrix be decomposed into a more succinct manner?

Let $A_m$ and $B_{ij}$ be matrices of size $M\times N$ where $m\in\{1,2\}$ and $N \times N$ where $i,j\in\{1,2\}$, respectively. Note that $m,i,j$ can be any number in general, but here they are specified so that the question can be clearer and easy to see.
I am wondering whether there is an alternative way to write/decompose the following block matrix more succinctly. I have had a look at Khatri-Rao product or Tracy-Singh product, but neither of them fits into the block matrix I have. I'm looking for a kind of block-wise multiplication of block matrix although I am not sure whether this type of matrix multiplication exists.
The matrix what I have is as follows:
$$
\begin{bmatrix}
A_1B_{11}A_1^T & A_1B_{12}A_2^T\\
A_2B_{21}A_1^T & A_2B_{22}A_2^T
\end{bmatrix}
$$
I would be so grateful if anyone can suggest how this matrix can be decomposed or written more succinctly.

A:

One way is
$$\begin{bmatrix}
A_1B_{11}A_1^T & A_1B_{12}A_2^T\\
A_2B_{21}A_1^T & A_2B_{22}A_2^T\end{bmatrix}
=
\begin{bmatrix}
A_1 & 0\\
0 & A_2
\end{bmatrix}
\begin{bmatrix}
B_{11} & B_{12}\\
B_{21} & B_{22}
\end{bmatrix}
\begin{bmatrix}
A_1 & 0\\
0 & A_2
\end{bmatrix}^T\,.$$

