# Are Wolf spaces flag manifolds?

It's all in the title: Are Wolf spaces flag manifolds? Both are group quotients of semi-simple Lie groups. In the Grassmannian case this is so, and I always tacitly assumed it extended to the general case. However, I recently read this post by Moroianu, where he says that

the quaternionic projective spaces $ℍP_n$ are quaternion-Kähler, but have no almost complex structure

Flag manifolds have the form $G/C(S)$ where $C(S)$ is the centralizer (in $G$) of its center $S$ (a torus).
Most $G/H$ in the list don't have this form, for $H$ has discrete center in each case except the complex Grassmannians $\mathrm{SU}(p+2)/\mathrm S(\mathrm{U}(p)\times\mathrm{U}(2))$ and $\mathrm{SO}(6)/\mathrm{SO}(2)\mathrm{SO}(4)$.
$$G/L\cdot S^1 \to G/L\cdot Sp(1)$$