First Laplace eigenvalue of strongly isotropy irreducible spaces

We study the smallest positive eigenvalue ${\lambda }_1$ of the Laplace-Beltrami operator associated with any compact strongly isotropy irreducible space. We provide an explicit expression for all simply connected cases. Furthermore, every strongly isotropy irreducible space is automatically an Einstein manifold, and we prove for each of them that $E<{\lambda }_1\le 16E$, where $E$ denotes the corresponding Einstein constant.