  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 $SO(5)/SO(2)\times SO(2) \times
SO(1)$ what are the conditions for a metric adapted to the
$\mathbb{Z}_2^2$-symmetric structure to be naturally reductive.
