  In this article we study the stability problem for the Einstein-Hilbert
functional on compact symmetric spaces following and completing the seminal
work of Koiso on the subject. We classify in detail the irreducible
representations of simple Lie algebras with Casimir eigenvalue less than the
Casimir eigenvalue of the adjoint representation, and use this information to
prove the stability of the Einstein metrics on both the quaternionic and Cayley
projective plane. Moreover we prove that the Einstein metrics on quaternionic
Grassmannians different from projective spaces are unstable.
