The paper is devoted to an abstract axiomatic version of a construction of boundary triplets implicit in the works of M.I. Vishik and G. Grubb and its applications to the index of families of self-adjoint elliptic differential boundary problems of order one. This leads to an analytic proof of the index theorem for Dirac-like self-adjoint boundary problems from arXiv:2207.09574, and to an Agranovich-Dynin type theorem computing the difference of indices of families of self-adjoint boundary problems differing only by the boundary conditions.