On every generalized flag manifold, an invariant almost Hermitian pair (g,J) is G1 exactly when the metric parameters are equal on every triple of t-roots with equal signs and zero sum, and the normal metric is the unique metric that is G1 for every invariant almost complex structure.
Arvanitoyeorgos, An Introduction to Lie Groups and the Geometry of Homogeneous Spaces, STML 22 Amer
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G1 structures on flag manifolds
On every generalized flag manifold, an invariant almost Hermitian pair (g,J) is G1 exactly when the metric parameters are equal on every triple of t-roots with equal signs and zero sum, and the normal metric is the unique metric that is G1 for every invariant almost complex structure.