In Weibel's *An Introduction to Homological Algebra*, the Chevalley-Eilenberg complex of a Lie algebra $g$ is defined as $\Lambda^*(g) \otimes Ug$ where $Ug$ is the universal enveloping algebra of $g$. The differential here has degree -1.

I have been told that the Chevalley-Eilenberg complex for $g$ is $C^*(g) = \text{Sym}(g^*[-1])$, the free graded commutative algebra on the vector space dual of $g$ placed in degree 1. The bracket $[,]$ is a map $\Lambda^2 g \to g$ so its dual $d : = [,]* \colon g^* \to \Lambda^2 g^*$ is a map from $C^1(g) \to C^2(g)$. Since $C^*(g)$ is free, this defines a derivation, also called $d$, from $C^*(g)$ to itself. This derivation satisfies $d^2 = 0$ precisely because $[,]$ satisfies the Jacobi identity.

Finally, Kontsevich and Soibelman in *Deformation Theory I* leave it as an exercise to construct the Chevalley-Eilenberg complex in analogy to the way that the Hochschild complex is constructed for an associative algebra by considering formal deformations.

The first is a chain complex, the second a cochain complex, and what do either have to do with formal deformations of $g$?