Despite use of blended cements containing significant amounts of aluminum for over 30 years, the structural nature of aluminum in the main hydration product, calcium aluminate silicate hydrate (C-A-S-H), remains elusive. Using first-principles calculations, we predict that aluminum is incorporated into the bridging sites of the linear silicate chains and that at high Ca:Si and H2O ratios, the stable coordination number of aluminum is six. Specifically, we predict that silicate-bridging [AlO2(OH)4]5- complexes are favored, stabilized by hydroxyl ligands and charge balancing calcium ions in the interlayer space. This structure is then confirmed experimentally by one- and two-dimensional dynamic nuclear polarization enhanced 27Al and 29Si solid-state NMR experiments. We notably assign a narrow 27Al NMR signal at 5 ppm to the silicate-bridging [AlO2(OH)4]5- sites and show that this signal correlates to 29Si NMR signals from silicates in C-A-S-H, conflicting with its conventional assignment to a "third aluminate hydrate" (TAH) phase. We therefore conclude that TAH does not exist. This resolves a long-standing dilemma about the location and nature of the six-fold-coordinated aluminum observed by 27Al NMR in C-A-S-H samples.