  We remark that on a compact inner symmetric space $G/K$, indowed with the
Riemmannian metric given by the Killing form of $G$ signed-changed, the first
(non-zero) eigenvalue of the Laplace operator on $1$-forms is the Casimir
eigenvalue of the highest either long or short root of $G$, according as the
highest weight of the isotropy representation is long or short. Some results
for the first (non-zero) eigenvalue on functions are derived. This is a
revision of the first version of the preprint: a non correct statement about
the spectrum on functions has been reviewed.
