The index of exceptional symmetric spaces

The index of a Riemannian symmetric space is the minimal codimension of a proper totally geodesic submanifold. There is a conjecture for how to calculate the index. In this paper we give an affirmative answer to this conjecture for the exceptional Riemannian symmetric spaces and for the classical symmetric spaces Sp${}_r(\mathbb{R})/U_r$. Our methodology is new and based on the idea of using slice representations for studying totally geodesic submanifolds.