First, I consider $I$ and $J$ as the rows and columns that do appear in $A_{I,J}$. Thus, the maximal sets in your example are the empty ones.

Note that this is not just a sequence of minors. They are: (1) nested; (2) "symmetric": the indices of the columns and the rows coincide; (3) "central": the diagonal of the minor lies on the diagonal of the matrix. I think that I took care enough to avoid this confusion.