I would like to illustrate the mathematical operation of the direct sum of matrices in the case where the matrices are not appended one after the other along the diagonal, but instead mixed among one another. This must be generalizable to any permutation of mixing indices along the diagonal. This is for the purpose of plugging this into Mathematica, but also illustrating the exact operation clearly on paper.
For example, say we have matrices $A$, $B$ and $C$ \begin{align} A=& \begin{bmatrix} \cos (t) & \sin(t) \\ -\sin(t) & \cos(t) \end{bmatrix}\\[10pt] B=& \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\\[10pt] C=& \begin{bmatrix} \cos (u) & \sin(u) \\ -\sin(u) & \cos(u) \end{bmatrix}\\[10pt] A ? B ? C=& \begin{bmatrix} \cos (t) & \sin(t) & 0 & 0 & 0 & 0\\ -\sin(t) & \cos(t) & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & \cos(u) & 0 & \sin(u)\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -\sin(u) & 0 & \cos(u) \end{bmatrix} \end{align}
How might this be written and what is the mathematical operation "?".