Let { α t : t ∈ R } and { β t : t ∈ R } be two commuting one-parameter groups of ∗ -automorphisms of a von Neumann algebra M such that α t + α − t = β t + β − t for all t ∈ R . The purpose of this note is to provide a simple and short proof of the central decomposition result: α t = β t on M p and a α t = β − t on M ( 1 − p ) for a central projection p ∈ M , without using the theory of spectral subspaces.