  The search for the chiral magnetic effect (CME) in relativistic heavy-ion
collisions helps us understand the $\mathcal{CP}$ symmetry breaking in strong
interactions and the topological nature of the QCD vacuum. Since the background
and signal of the CME have different correlations with the spectator and
participant planes, a two-plane method has been proposed to extract the
fraction of the CME signal inside the CME observable of $\Delta\gamma$ from the
experimental measurements relative to the two planes. Using a multiphase
transport model with different strengths of the CME, we reexamine the two-plane
method in isobar collisions at $\sqrt{s_{_{\rm NN}}} = 200$ GeV. The ratios of
the CME signals and the elliptic flow backgrounds relative to the two different
planes are found to be different, which is inconsistent with the assumptions
made in the current experimental measurements. This difference arises from the
decorrelation effect of the chiral magnetic effect relative to the spectator
and participant planes caused by final state interactions. Our finding suggests
that the current experimental measurements may overestimate the fraction of the
CME signal in the CME observable in the final state of relativistic heavy-ion
collisions.
