  We propose an algebraic analysis using a 3+1 decomposition to identify
conditions for a clever cancellation of the higher derivatives, which plagued
the theory with Ostrogradsky ghosts, by exploiting some existing degeneracy in
the Lagrangian. We obtain these conditions as linear equations (in terms of
coefficients of the higher derivative terms) and demand that they vanish, such
that the existence of nontrivial solutions implies that the theory is
degenerate. We find that, for the theory under consideration, no such solutions
exist for a general inhomogeneous scalar field, but that the theory is
degenerate in the unitary gauge. We, then, find modified FLRW equations and
narrow down conditions for which there could exist a de Sitter inflationary
epoch. We further find constraints on the coefficients of the remaining
higher-derivative interaction terms, based on power-counting renormalizability
and tree-level unitarity up to the Planck scale.
