Fix a base ring $k$ and $k$-Lie algebras $\mathfrak{s}$ and $\mathfrak{t}$, and consider the slice category $\mathfrak{s}/\mathrm{Lie}_k/\mathfrak{t}$. This is a category of universal algebras, so there's a Quillen cohomology theory for its objects; in short, $$H^n_{\mathfrak{s}/\mathrm{Lie_k}/\mathfrak{t}}(\mathfrak{g};\mathfrak{a}) = \pi_0 \mathbb{R}\mathrm{Hom}(\mathfrak{g},B^n \mathfrak{a}),$$ defined using simplicial resolutions.
Question: What are the above groups concretely (e.g. in terms of suitable $\mathrm{Ext}$ groups over some universal enveloping algebra)?
Quillen mentions a bit about Lie algebras in his "On the (co)-homology of commutative rings", but it's a little sparse on details, and I'm not sure what happens for these general slice categories. And there are plenty of sources for "Lie algebra cohomology", only its precise relation to Quillen cohomology is not entirely clear to me. If there's a source that makes this clear, it would probably answer my question.
Here's an additional, roughly equivalent, question. We can identify $\mathrm{Ab}(\mathfrak{s}/\mathrm{Lie_k}/\mathfrak{t}) = \mathrm{LMod}_{U(\mathfrak{t})}$ as a category of $\mathfrak{t}$-modules. If $\mathfrak{s} = \mathfrak{t}$, then I think abelianization works as follows: given $\mathfrak{g}\in\mathfrak{t}/\mathrm{Lie_k}/\mathfrak{t}$, we can additively split $\mathfrak{g} = \mathfrak{t}\oplus\mathfrak{i}$, and we send this to $D(\mathfrak{g}) := \mathfrak{i}/[\mathfrak{i},\mathfrak{i}]$. And I guess the general case reduces to understanding this case and what happens with pushouts of Lie algebras. This leads to
Question: Is there a more explicit description of $\mathbb{L}D(\mathfrak{g})$? (e.g. is it something like $U(\mathfrak{t})\otimes_{U(\mathfrak{g})}^{\mathbb{L}} U(\mathfrak{t})$?)
Of course if there's a direct description of derived abelianization for $\mathfrak{s}/\mathrm{Lie_k}/\mathfrak{t}$ without first reducing to $\mathfrak{s}=\mathfrak{t}$, that's even better.
If it's necessary to restrict $k$ to be a field for a nice description, that's fine, although I'd prefer to allow general $k$ and just require various flatness or projectivity assumptions wherever it's necessary for a clean description. However I'd like to allow $k = \mathbb{F}_p$. My understanding is that for the universal enveloping algebra to be well behaved we must enforce that our Lie algebras satisfy $[x,x]=0$ and $[x,[x,x]]=0$; these are of course automatic if $6$ is invertible no matter how you define your Lie algebras. I'd be happy whether an answer assumes this or not, just please specify the definition you take.
