  The lagrangian formalism for the supermembrane in any 11d supergravity
background is constructed in the pure spinor framework. Our gauge-fixed action
is manifestly BRST, supersymmetric, and 3d Lorentz invariant. The relation
between the Free Differential Algebras (FDA) underlying 11d supergravity and
the BRST symmetry of the membrane action is exploited. The "gauge-fixing" has a
natural interpretation as the variation of the Chevalley cohomology class
needed for the extension of 11d super-Poincare' superalgebra to M-theory FDA.
We study the solution of the pure spinor constraints in full detail.
