To start, I would like to note that my background on Lie algebras is quite basic, so this question might be trivial when seen from a Lie algebra perspective, which I lack.
We have the concept of a reductive homogeneous space $G/H$, which is one for which the Lie algebra of $G$ splits as $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$ that is $\operatorname{Ad}(H)$ invariant, $\operatorname{Ad}_h(\mathfrak{m}) \subseteq \mathfrak{m}$. This structure gives a left-invariant connection on $G$ that descends into a left-invariant connection on $G/H$, that is, it is a principal connection on $\pi : G \to G/H$.
Now, there is this other concept of a Lie algebra being reductive if it's a direct sum of a semi-simple Lie algebra and an Abelian Lie algebra. I get the feeling that this concept is somewhat related to the concept above, as for (for example real) matrices, for a connected closed subgroup $G \leq \operatorname{GL}(n)$ closed under the transpose, its algebra is reductive, and I believe that you can consider $H = G \cap \operatorname{Ort}(n)$ so that $G/H$ is a reductive space taking $\mathfrak{m} = \mathfrak{g}\cap\operatorname{Sym}(n)$, and by the polar decomposition $G/H \cong \exp(\mathfrak{m})$.
My questions are:
Is there any other relation between reductive Lie algebras (and Lie groups) and reductive spaces besides this?
Is there a way to, given a reductive Lie algebra, to construct a reductive homogeneous space?
If we take a reductive Lie group $G$ and a maximal compact subgroup $H$ of $G$, is it true that $G/H$ is a reductive homogeneous space?
Any reference that covers ideas related to these would be highly appreciated!
This is a repost from an mathstackexchange question
