See also reductive Cartan geometry.
The same as reductive Cartan geometry.
Question for MO of myself here. As a conclusion:
"So a reductive homogeneous space, in the sense of Sharpe's book, is precisely a homogeneous space with invariant connection on the bundle $G\to G/H$, i.e. precisely a homogeneous space with an $H$-module decomposition $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{p}$."
"The choice of $\mathfrak{p}$ is not given a priori: rather, every $G$-invariant connection on $M = G/H$ determines one such $\mathfrak{p}$ and vice versa, so the choices of $\mathfrak{p}$ are arbitrary $\text{Ad}H$-invariant complements to $\mathfrak{h} \subseteq \mathfrak{g}$.""
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Author of the notes: Antonio J. Pan-Collantes
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