We show that for product sources, rate splitting is optimal for secret key agreement using limited one-way communication at two terminals. This yields an alternative proof of the tensorization property of a strong data processing inequality originally studied by Erkip and Cover and amended recently by Anantharam et al. We derive a `water-filling' solution of the communication-rate--key-rate tradeoff for two arbitrarily correlated vector Gaussian sources, for the case with an eavesdropper, and for stationary Gaussian processes.