Pontryagin classes of locally symmetric manifolds

Pontryagin classes p(i)(M) are basic invariants of a smooth manifold M, and many topological problems can be reduced to computing these classes. For a locally symmetric manifold, Borel and Hirzebruch gave an algorithm to determine if p(i)(M) is nonzero. In addition they implemented their algorithm for a few well-known M and for i = 1, 2. Nevertheless, there remained several M for which their algorithm was not implemented. In this note we compute low-degree Pontryagin classes for every closed, locally symmetric manifold of noncompact type. As a result of this computation, we answer the question: Which closed locally symmetric M have at least one nonzero Pontryagin class?