Metric on Homogeneous Space $G/H$ For simplicity, assume $G$ is compact and semi-simple Lie group, and $H$ is a closed subgroup of $G$. Therefore the homogeneous space is reductvie, say $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ where $\mathfrak{g}$, $\mathfrak{h}$ and $\mathfrak{m}$ are Lie algebra of $G$, $H$, and complimentary space to $H$.
I know there is a fundamental theorem on the metric $\gamma$ of $G/H$ that “there is a one-to-one correspondence between the $G$-invariant metric $\gamma$ on $G/H$ and the ${\rm ad}_\mathfrak{h}$-invariant inner product $\eta$ on $\mathfrak{m}$”.
For semi-simple and compact Lie group, one can always find negative definite Killing form $K$ on $\mathfrak{g}$, which is apparently ${\rm ad}_\mathfrak{h}$-invariant. My question is that, given the Killing form $K$, how many $\mathfrak{m}$ compliment to $\mathfrak{h}$ one can choose, so that $\mathfrak{m}$ is ${\rm ad}_\mathfrak{h}$ invariant. I know that for example one can choose the orthogonal $\mathfrak{m}=\mathfrak{h}^\perp$ by Killing form $K$. Is there any options and if so is there any relation among those metric $\gamma$ or $\eta$ induced by the same Killing form $K$
Thanks very much
 A: $\mathfrak{h}$-invariant complements to $\mathfrak{h}\subseteq\mathfrak{g}$ correspond to splittings of the canonical short exact sequence of $\mathfrak{h}$-modules:
$$0\to \mathfrak{h}\to\mathfrak{g}\to\mathfrak{g}/\mathfrak{h}\to 0.$$
Let $s_0:\mathfrak{g}/\mathfrak{h}\to \mathfrak{g}$ denote the splitting whose image is $\mathfrak{h}^\perp$.
Given a second splitting, $s_1:\mathfrak{g}/\mathfrak{h}\to\mathfrak{g}$, their difference $s_1-s_0:\mathfrak{g}/\mathfrak{h}\to\mathfrak{g}$ factors through $\mathfrak{h}$, i.e. can be identified with an element $\sigma=(s_1-s_0)\in\operatorname{Hom}_{\mathfrak{h}}(\mathfrak{g}/\mathfrak{h},\mathfrak{h})\cong \operatorname{Hom}_{\mathfrak{h}}(\mathfrak{h}^\perp,\mathfrak{h})$.
The metric on $T_HG/H\cong \mathfrak{g}/\mathfrak{h}$ induced by the spliting $s_1$ and the killing form $K$ is just the pullback $s_1^*K$, i.e. for $\xi,\eta\in \mathfrak{g}/\mathfrak{h}$, the metric evaluates to 
$$s_1^*K(\xi,\eta)=K(s_1(\xi),s_1(\eta))=K(s_0(\xi)+\sigma(\xi),s_0(\eta)+\sigma(\eta))=K(s_0(\xi)),s_0(\eta))+K(\sigma(\xi),\sigma(\eta))=(s_0^*K+\sigma^*K)(\xi,\eta),$$
where we have used the fact that the images of $s_0$ and $\sigma$ are orthogonal.
So, to summarize, the space of $\mathfrak{h}$-invariant complements can be identified with $\operatorname{Hom}_{\mathfrak{h}}(\mathfrak{h}^\perp,\mathfrak{h})$, and for any $\sigma\in\operatorname{Hom}_{\mathfrak{h}}(\mathfrak{h}^\perp,\mathfrak{h})$, the corresponding metric is modified by adding $\sigma^*K$ to it. 
