The main question here is to ask if anyone has ever seen/researched the map $\alpha$ below and get a reference regarding it. Also, if anyone tells me the equivalent conditions to $\alpha=0$, I would be very appreciated.
The map is $$\alpha: \Lambda^i \mathfrak g^\vee \otimes \mathfrak h/\phi(\mathfrak g) \rightarrow \Lambda^{i+2} \mathfrak g^\vee \otimes \ker \phi$$ of the form
$$\alpha(f\otimes c) = x^a x^b f \otimes p\left ([x_a,\psi([\phi(x_b),j(c)]_{\mathfrak g})]_{\mathfrak h}\right )$$
for a Lie algebra morphism $\phi:\mathfrak g \rightarrow \mathfrak h$, a vector space morphism $\psi:\mathfrak h \rightarrow \mathfrak g$ satisfying $\phi\circ \psi \circ \phi = \phi$ and $\psi \circ \phi \circ \psi = \psi$, $p:\mathfrak g \rightarrow \ker \phi$ and $j: \mathfrak h/\phi(\mathfrak g)\rightarrow \mathfrak h$. This can also be explained in terms of differentials of Chevalley Eilenberg differentials (instead of Lie brackets). The detailed explanations are below.
Let $\mathfrak g$ and $\mathfrak h$ be two finite-dimensional Lie algebras over a field of characteristic 0. Given a Lie algebra morphism $\phi:\mathfrak g \rightarrow \mathfrak h$, the Lie algebra $\mathfrak h$ is a $\mathfrak g$-module. Therefore we get an exact sequence of vector spaces
$$ 0\rightarrow \mathfrak k \xrightarrow{\iota} \mathfrak g \xrightarrow{\phi} \mathfrak h \xrightarrow{q} \mathfrak c \rightarrow 0$$
where $\mathfrak k = \ker \phi$ and $\mathfrak c = \mathfrak h / \phi(\mathfrak g)$ (and $\mathfrak c$ is not a Lie algebra in general).
Consider the vector space splitting of the exact sequence: $p:\mathfrak g\rightarrow \mathfrak k$ and $j:\mathfrak c \rightarrow \mathfrak h$ satisfying $p\circ \iota=1_{\mathfrak k}$ and $q \circ j = 1_{\mathfrak c}$ induces $\psi:\mathfrak h \rightarrow \mathfrak g$ such that $\phi\circ \psi \circ \psi = \phi$ and $\psi\circ \phi \circ \psi = \psi$.
It is clear that $\mathfrak k$, $\mathfrak g$ and $\mathfrak h$ are $\mathfrak g$-module via the adjoint action (and with $\phi$ for $\mathfrak h$) and we can further define a $\mathfrak g$-module structure on $\mathfrak c$ induced by the $\mathfrak g$-module structure on $\mathfrak h$.
Now consider the Chevalley-Eilenberg complexes of those $\mathfrak g$-modules. Now we have
$$0\rightarrow \Lambda^\bullet \mathfrak g^\vee \otimes \mathfrak k \xrightarrow{1\otimes \iota} \Lambda^\bullet \mathfrak g^\vee \otimes \mathfrak g \xrightarrow{1\otimes \phi} \Lambda^\bullet \mathfrak g^\vee \otimes \mathfrak h \xrightarrow{1\otimes q} \Lambda^\bullet \mathfrak g^\vee \otimes \mathfrak c \rightarrow 0$$
which is an exact sequence of chain complexes.
From here I could make a map from $\alpha:\Lambda^i \mathfrak g^\vee \otimes \mathfrak c \rightarrow \Lambda^{i+2} \mathfrak g^\vee \otimes \mathfrak k$ by defining
$$\alpha(f\otimes c) = x^a x^b f \otimes p\left ([x_a,\psi([\phi(x_b),j(c)]_{\mathfrak g})]_{\mathfrak h}\right )$$
where $f\in \Lambda^i \mathfrak g^\vee$, $c\in \mathfrak c$, $x^a, x^b \in \mathfrak g^\vee$ and $x_a, x_b \in \mathfrak g$ in that $x^a$ and $x_a$ are dual to each other (and the same for $x^b$ and $x_b$).
With the standard Chevalley-Eilenberg differentials on $\mathfrak g$-modules $\mathfrak g$ and $\mathfrak h$, this map $\alpha$ can be written in the following form (up to sign)
$$\iota \circ \alpha(f\otimes c) = (d_{CE}^\mathfrak g \circ \psi - \psi \circ d_{CE}^\mathfrak h)\circ \phi \circ (d_{CE}^\mathfrak g \circ \psi - \psi \circ d_{CE}^\mathfrak h) (f\otimes j(c)).$$
You can also extend this $\alpha$ to be defined on $\Lambda^\bullet \mathfrak g^\vee \otimes \mathfrak h$ simply by
$$\iota \circ \bar \alpha (f\otimes h)=(d_{CE}^\mathfrak g \circ \psi - \psi \circ d_{CE}^\mathfrak h)\circ \phi \circ (d_{CE}^\mathfrak g \circ \psi - \psi \circ d_{CE}^\mathfrak h)(f\otimes h).$$
It is clear that if $\phi$ is either injective or surjective, $\alpha=0$, but I could not figure out what would be the complete conditions for $\alpha$ to be 0. If there are any obstructions on some sort of cohomology that tells us about this $\alpha$ to be 0 or anything that is related to this $\alpha$, please share it with me.
Additional explanation using dg-manifolds:
If you know dg-manifolds, then $\mathfrak g[1]$ is a dg-manifold. Then all those things can be understood as a differential on the section of dg-vector bundle $$\mathfrak g[1]\times (\mathfrak k[1] \oplus \mathfrak c) \rightarrow \mathfrak g[1]$$ over the manifold $\mathfrak g[1]$. (This dg-vector bundle is related to a vector bundle $T_{\mathfrak g[1]}\times \mathfrak h \rightarrow \mathfrak g[1]$.) Then the $\alpha$, with the corresponding degree shifting, defines a (non-standard) differential on its section.
