I've calculated a particular invariant 3-form on a reductive homogeneous space $G/H$ that is associated to an invariant principal connection $A$ on $G\to G/H$, and I want to know if it is something that appears in the literature. I'm not familiar with all the various specialty tensors that turn up in this space, and it's quite tricky to search for a specific one.
Write $\mathfrak{g} = \mathfrak{h}\oplus \mathfrak{m}$, for $\mathfrak{m}\simeq T_{eH}G/H$ the Ad-invariant horizontal subspace for $A$ at the identity (using reductivity here). Let $F_A$ be the curvature 2-form (on $G$) of $A$, $\Theta$ be the Maurer–Cartan form on $G$, and I am assuming $G$ is compact simple, with the negative of the Killing form on $\mathfrak{g}$ denoted by $\langle-,-\rangle$.
The 3-form is then this: $$ \langle \Theta, F_A\rangle\big|_{\mathfrak{m}}. $$ The restriction abuses notation somewhat and is meant to indicate that one only evaluates this on tangent vectors (that when translated back to the identity are) in $\mathfrak{m}$. There is a formula on Lie algebra elements one can write down that involves ternary forms like $\langle X,[Y,Z]_{\mathfrak{h}}\rangle$, for $X,Y,Z\in\mathfrak{m}$.
What is this 3-form? In the case that $\mathfrak{m} \perp \mathfrak{h}$ with respect to the inner product, then this vanishes, so it measures the deformation, as it were, of $A$ from that canonical connection. This feels like something that should be known. In particular, it feels reminiscent of the results that classify homogeneous connections via appropriate maps $\mathfrak{m}\to \mathfrak{gl}(\mathfrak{m})$, but it's not using a modified bracket $[-,-]_{\mathfrak{m}}$, as you would need.
