  Construction of the gauge-invariant variables for the linear metric
perturbation, which was proposed in the paper [K. Nakamura, arXiv:1101.1147],
is discussed through an alternative approach. Our starting point of the
construction of the gauge-invariant variables is an non-trivial non-local
decomposition of the linear metric perturbation. Assuming the existence of some
Green functions, we reproduce results in the above paper. This supports the
consistency of the result and implies that one can develop the
general-relativistic higher-order gauge-invariant perturbation theory on
general background spacetime.
