  A suitable generalisation of the Lichnerowicz formula can relate the squares
of supersymmetric operators to the effective action, the Bianchi identities for
fluxes, and some equations of motion. Recently, such formulae have also been
shown to underlie the (generalised) geometry of supersymmetric theories. In
this paper, we derive an M-theoretic Lichnerowicz formula that describes
eleven-dimensional supergravity together with its higher-derivative couplings.
The first corrections to the action appear at eight-derivative level, and the
construction yields two different supersymmetric invariants, each with a free
coefficient. We discuss the restriction of our construction to
seven-dimensional internal spaces, and implications for compactifications on
manifolds of $G_2$ holonomy. Inclusion of fluxes and computation of
contributions with higher than eight derivatives are also discussed.
