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.
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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.