The derivation of the time-dependent variational equations of the Multi-Configuration Time-Dependent Hartree (MCTDH) method for high-dimensional quantum propagation is revisited from the perspective of tangent space projection methods. In this context, we focus on a recently introduced algorithm [C. Lubich, Appl... Math. Res. eXpress 2015, 311 (2015), B. Kloss et al., J. Chem. Phys. 146, 174107 (2017)] for the integration of the MCTDH equations, which relies on a suitable splitting of the tangent space projection. The new integrator circumvents the direct inversion of reduced density matrices that appears in the standard method, by employing an auxiliary set of non-orthogonal single-particle functions. Here, we formulate the new algorithm and the underlying alternative form of the MCTDH equations in conventional chemical physics notation, in a complementary fashion to the tensor formalism used in the original work. Further, key features of the integration scheme are highlighted. (read more)