  Let $G/K$ be a simply connected spin compact inner irreducible symmetric
space, endowed with the metric induced by the Killing form of $G$ sign-changed.
We give a formula for the square of the first eigenvalue of the Dirac operator
in terms of a root system of $G$. As an example of application, we give the
list of the first eigenvalues for the spin compact irreducible symmetric spaces
endowed with a quaternion-K\"{a}hler structure.
