A Note on Homogeneous Geometry of \(G_2\) -Type Real Flag Manifolds

In this note, we describe the invariant metrics on real flag manifolds of the split real form of the exceptional Lie group \(G_2\) , starting from the decomposition of the isotropy representation into irreducible invariant subspaces. This description establishes the basic pillars for the study of Riemannian geometry and homogeneous geometric flows on these manifolds.