Invariant Einstein metrics on flag manifolds with four isotropy summands

A generalized flag manifold is a homogeneous space of the form $G\mathrm{/}K$, where $K$ is the centralizer of a torus in a compact connected semisimple Lie group $G$. We classify all flag manifolds with four isotropy summands and we study their geometry. We present new $G$-invariant Einstein metrics by solving explicity the Einstein equation. We also examine the isometric problem for these Einstein metrics.