  We construct a Lie-Rinehart algebra over an infinitesimal extension of the
space of initial value fields for Einstein's equations. The bracket relations
in this algebra are precisely those of the constraints for the initial value
problem. The Lie-Rinehart algebra comes from a slight generalization of a Lie
algebroid in which the algebra consists of sections of a sheaf rather than a
vector bundle. (An actual Lie algebroid had been previously constructed by
Blohmann, Fernandes, and Weinstein over a much larger extension.) The
construction uses the BV-BFV (Batalin-Fradkin-Vilkovisky) approach to boundary
value problems, starting with the Einstein equations themselves, to construct
an $L_\infty$-algebroid over a graded manifold which extends the initial data.
The Lie-Rinehart algebra is then constructed by a change of variables. One of
the consequences of the BV-BFV approach is a proof that the coisotropic
property of the constraint set follows from the invariance of the Einstein
equations under space-time diffeomorphisms.
