Solved – Difference between “common index” and “no common index”

From the documentation provided by whuber, it seems that 'index' refers to the indices of $rho$. In the no-common case:

26.3.1 General case: No common index

We want to perform an a priori analysis for a one-sided test whether $rho_{1,4} = rho_{2,3}$ or $rho_{1,4} < rho_{2,3}$ holds.

And in the common case:

26.3.2 Special case: Common index

Assuming again the population correlation matrix $C_p$ shown above, we want to do an a priori analysis for the test whether $rho_{1,3} = rho_{2,3}$ or $rho_{1,} < rho_{2,3}$ holds. … we have the following identities: $a = 3$ (the common index) …

Index in this context refers to the indices of the correlation matrix. The item indexed at $i,j$ gives the correlation between variables $i$ and $j$.

To give a simple example, if you were testing the correlation between height and weight and the correlation between height and arm circumference, they share a common index. If you were comparing the correlations between height, weight and between arm circumference, leg circumference, they do not share a common index.

(Apologies for not using your example; the variables to correlate were unclear to me.)