In my matrix algebra book, Matrix Algebra Useful For Statistics by S. Searle and A. Khuri, I came across several equations that include upper and lower case indices for rows and columns. This dichotomy confuses me, and I have not seen this kind of notation anywhere else except in this book. Unfortunately, there is no explanation for this notation. What reason is there to distinguish between lower and upper case letters?
Here is an example of an equation for differentiating determinants of symmetric matrices that uses this notation:
$$\dfrac{\partial|\boldsymbol{X}|}{\partial x_{IJ}} = |\boldsymbol{X}_{iJ}| + |\boldsymbol{X}_{IJ}| - \delta_{ij}|\boldsymbol{X}_{Ij}| \quad \text{for symmetric } \boldsymbol{X}$$
where $|\boldsymbol{X}_{ij}|$ is a minor, i.e. the determinant of $|\boldsymbol{X}|$ with its $i$th row and $j$th column removed.
Update: Another example, which uses this notation, is the definition of the determinant expansion by minors (expansion by rows)
$$|\boldsymbol{A}| = \sum_{J=1}^n a_{ij}(-1)^{I+j}|\boldsymbol{M}_{ij}| \quad \text{for any } i$$
which clearly shows that all $i$'s, whether lower or upper case, are identical. Another point that bothers me here is that the index of the summation is an upper case $J$, even though all the column indices in the equation are lower case.
At first, I thought it might be because the authors wanted to emphasize the index of the expansion, but that cannot be the case since it is not done consistently. Especially considering that the same definition when expanding the elements of the column of some matrix $\boldsymbol{A}$ is,
$$|\boldsymbol{A}| = \sum_{I=1}^n a_{IJ}(-1)^{I+j}|\boldsymbol{M}_{Ij}| \quad \text{for any } j$$
Even though it is not a very crucial question, I still wonder what the meaning of this notation might be, since I do not see its relevance.