Einstein Homogeneous Bisymmetric Fibrations

We consider a homogeneous fibration $G\mathrm{/}L\to G\mathrm{/}K$, with symmetric fiber and base, where $G$ is a compact connected semisimple Lie group and $L$ has maximal rank in $G$. We suppose the base space $G\mathrm{/}K$ is isotropy irreducible and the fiber $K\mathrm{/}L$ is simply connected. We investigate the existence of $G$-invariant Einstein metrics on $G\mathrm{/}L$ such that the natural projection onto $G\mathrm{/}K$ is a Riemannian submersion with totally geodesic fibers. These spaces are divided in two types: the fiber $K\mathrm{/}L$ is isotropy irreducible or is the product of two irreducible symmetric spaces. We classify all the $G$-invariant Einstein metrics with totally geodesic fibers for the first type. For the second type, we classify all these metrics when $G$ is an exceptional Lie group. If $G$ is a classical Lie group we classify all such metrics which are the orthogonal sum of the normal metrics on the fiber and on the base or such that the restriction to the fiber is also Einstein.