Riemannian symmetries in flag manifolds

Flag manifolds are in general not symmetric spaces. But they are provided with a structure of ${\mathbb{Z}}_2^k$-symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. We detail for the flag manifold $SO(5)/SO(2)\times SO(2) \times SO(1)$whataretheconditionsforametricadaptedtothe$$\mathbb{Z}$_2^2$-symmetric structure to be naturally reductive.