Forgive me if this question is too elementary however, I haven't found an answer. If $\mathfrak{g}$ is a Lie algebra one can define its Lie algebra cohomology: the definition is quite similar to the way in which de Rham cohomology is defined. For such theory we can consider coefficients in arbitrary $\mathfrak{g}$-module$ M$. If $\mathfrak{h} \subset \mathfrak{g}$ is a Lie subalgebra one can define the relative theory using subcomplex consisting from cochains $\varphi$ satisfying $i_X \varphi=0$ and $i_X d\varphi =0$ for every $X \in \mathfrak{h}$. This indeed defines subcomplex and the cohomology of this complex is the relative cohomology of $\mathfrak{g}$ (relative to $\mathfrak{h}$).

How one can define cohomology of Lie algebra $\mathfrak{g}$ with respect to some group $H$?

Obviously I'm aware that $H$ must have something to do with $\mathfrak{g}$. I found one definition but I'm not sure whether I should cite it here since I'm not sure whether it makes sense (if I should, let me know in the comments and I will edit my post).

I would be very grateful if someone who is familiar with this topic could shed some light on it.

EDIT: The only definition which I saw is the following: we assume that $H$ is a Lie group such that $\mathfrak{h}:=Lie(H)$ acts on $\mathfrak{g}$ and on $M$ (the coefficient module) such that the differential of the action on $\mathfrak{g}$ is $ad_{\mathfrak{g}} \mathfrak{h}$. We consider the complex: $$C^{\bullet}(\mathfrak{g},H;M):=\{ \varphi \in Hom_H(\Lambda^{\bullet}\mathfrak{g},M): i_X \varphi =0 \ \ \forall_{X \in \mathfrak{h}}\}$$ and its cohomology. Maybe I should add that the particular case in which I'm interested is the case when $\mathfrak{g}$ is Lie algebra of all formal vector fields on $\mathbb{R}^n$ and $H=SO(n)$. More context in which I'm interested can be found in this post.