  In this paper, we describe and analyze the spectral properties of a symmetric
positive definite inexact block preconditioner for a class of symmetric, double
saddle-point linear systems.
  We develop a spectral analysis of the preconditioned matrix, showing that its
eigenvalues can be described in terms of the roots of a cubic polynomial with
real coefficients.
  We illustrate the efficiency of the proposed preconditioners, and verify the
theoretical bounds, in solving large-scale PDE-constrained optimization
problems.
