On Formality of Some Homogeneous Spaces

Let G/H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus TH of H is conjugate to a torus T beta whose Lie algebra t beta is the kernel of the maximal root beta of the root system of the complexified Lie algebra gc. We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernandez, Munoz, and Sanchez which contains a full analysis of the formality property of SO(3)-bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.