  This is the first of two papers in which we construct formal power series
solutions in external parameters to the vacuum Einstein equations, implementing
one bounce for the Belinskii-Khalatnikov-Lifshitz (BKL) proposal for spatially
inhomogeneous spacetimes. We use a graded Lie algebra, homological framework. A
dedicated filtration encodes key features of the BKL proposal, and we use it to
set up an unobstructed perturbative problem.
