The Gutzwiller projector technique has long been known as a method to include correlations in electronic structure calculations. We describe a model implementation for a Gutzwiller + LDA calculation in a localized-orbital restricted basis framework, emphasizing the protocol step by step and illustrating our specific procedure for this and future applications. We demonstrate the method with a classic problem, the ferromagnetism of bulk bcc Fe, whose nature is attracting fresh interest. In the conventional Stoner-Wohlfarth model, and in spin-polarized LDA calculations, the ferromagnetic ordering of iron sets in so that the electrons can reduce their mutual Coulomb repulsion, at the cost of some increase of electron kinetic energy. This balance may, however, be altered by correlations, which are strong for localized d orbitals. The present localized basis Gutzwiller + LDA calculation demonstrates how the ferromagnetic ordering of Fe may, in fact, entrain a decrease of kinetic energy at the cost of some increase of potential energy. This happens because, as foreshadowed long ago by Goodenough and others and more recently supported by LDA-DMFT calculations, correlations cause e(g) and t(2g) d orbitals to behave differently, with the weakly propagating e(g) states fully spin polarized and almost localized, and only t(2g) states forming a broad partly filled itinerant band. Owing to an intra-atomic Hund's rule exchange that aligns e(g) and t(2g) spins, the propagation of itinerant t(2g) holes is favored when different atomic spins are ferromagnetically aligned. This suggests a strong analogy with double exchange in iron ferromagnetism.