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We consider a general method for constructing dual Gabor elements different from the canonical dual. Our approach is based on combining two Gabor frames such that the generated frame-type operator Sg,γ is nonsingular. We provide necessary and sufficient conditions on the Gabor window functions g and γ such that Sg,γ is nonsingular for rational oversampling, considering both the continuous-time and the discrete-time settings. In contrast to the frame operator, the operator Sg,γ is, in general, not positive. Therefore, all results in Gabor analysis that are based on the positivity of the frame operator cannot be applied directly. The advantage of the proposed characterization is that the algebraic system for computing the Gabor dual elements preserves the high structure of usual Gabor frames, leading to computationally efficient algorithms. In particular, we consider examples in which both the condition number and the computational complexity in computing the proposed dual Gabor elements decrease in comparison to the canonical dual Gabor elements.