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

A Remark on the First Eigenvalue of the Laplace Operator on 1-forms for Compact Inner Symmetric Spaces