Finding the standard error for an estimate of the change in mean z-score

I have 2 normally distributed, standardized (mean = 0; SD = 1) measures of an outcome taken at 2 time points. To estimate the standard error for the mean change between the 2 timepoints, a colleague has suggested the following:

If the data are collected in a different sample at each timepoint, the variance of the mean change in standard deviation score (SDS) is 4/n, where n = sample size, and n/2 measurements are taken at each time point.

If the data are collected in the same sample at each time point, the variance of the mean change in SDS is ((4 * (1 - r)) / n), where r is the correlation between the 2 measures, and where n = sample size, and n/2 measurements are taken at each time point.

I can't get the calculation for the longitindal case to match their results when I try to replicate them; I can't get in touch with the person who has suggested these; and I'm not clever enought to figure out of they are simply incorrect, or if I am making a mistake somewhere.

Edit:

I can replicate the standard errors from my colleague with the following, but am not convinced this is correct, particuarly since the variance of the average of two standardized normal variables with correlation r should be (1/2) + (1/2 * r).

Longitudinal data: sqrt(4 * (1 - r)) / 2 * sqrt(n/2)

Cross-sectional data:
sqrt(4) / sqrt(n)