Let $k$ be a commutative ring and let $K$ be a commutative $k$-algebra. Let $k$-alg-$K$ be the category of $k$-algebras with $k$-algebra maps into $K$. Let $K$-mod be the category of $K$-modules. Let $M$ be a $K$-module.
Andre-Quillen cohomology studies the derived functor of the functor $\text{Der}_k(-, M):$ $k$-alg-$K$ $\rightarrow$ $K$-mod. This functor is naturally isomorphic to $\text{Hom}_K(-, M)$ composed with the functor $F:$ $k$-alg-$K$ $\rightarrow$ $K$-mod sending $A$ to $\Omega_{A/k} \otimes_A K$. The functor $F$ is called the abelianization functor. It is a sort of universal way of abelianizing $k$-alg-$K$ to get a setting for homology. So this follows the general intuition that homological algebra studies linearizations of categories (so that, e.g. $\pi_n$ turns into $H_n$).
In Akhil Mathew's post on the cotangent complex, he establishes the mentioned functor $F : k \text{-alg-} K \rightarrow K \text{-mod}$ sending $A$ to $\Omega_{A/k} \otimes_A K$ and a functor $U : K \text{-mod} \rightarrow k \text{-alg-} K$ sending $M$ to $M \oplus K$, where the multiplication on $M \oplus K$ sends $((x, a), (y, b))$ to $(xb + ay, ab)$. $F$ is left adjoint to $U$.
I am interested in the canonical cotriple resolution that $F$ and $U$ induce, which could also be called the bar resolution in Andre Quillen cohomology. Specifically, my question is, is $F \circ U$ naturally isomorphic to some simpler functor? Of course, we have $F \circ U (M) = \Omega_{M \oplus K / k} \otimes_{M \oplus K} K$. I want to be as explicit as possible, and I am particularly interested in an explicit description of the case where $K = k \oplus M$ for some module $M$.
Also, I am hoping it is related to the exterior algebra or symmetric algebra. I am hoping it is vaguely related to the De Rham complex.
Right now I am trying to show the following smaller claim: let $k$ be a field, and let $M$ be a $k$-vector space. Then $\Omega_{k \oplus M \oplus M / k} \otimes_{k \oplus M \oplus M} k \oplus M$ is isomorphic to either $\Lambda^2_k (M)$ or $S^2_k (M)$ (second grade of the symmetric algebra).
By the way, this cotriple plays the role that the cotriple resolution
$$\cdots \rightarrow k[k[k[G]]] \rightarrow k[k[G]] \rightarrow k[G]$$ plays in group cohomology.