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Let H be a Hilbert C * -module, and let H M be the indefinite inner space induced by a self-adjointable and invertible operator M on H . Given weighted projections P and Q on H M , let S λ,k = (P Q) k − λ(Q P) k for a pair (k, λ), where k is a natural number and λ is a complex number. It is proved that P Q − Q P is weighted Moore-Penrose invertible if and only if S λ,k is weighted Moore-Penrose invertible for every pair (k, λ).

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