I am trying to work out some examples of Maurer Cartan equation in deformation theory. For the associative case, I use the differential on the Hochschild complex and the Gerstenhaber bracket on $L_n = Hom_k(B^{\otimes(n+1)}, B)$, $B$ is an associative algebra, to get a DGLA structure on $L$. I then extend it to $L\otimes A$, $A = k[\epsilon]/\epsilon^4$ (I am trying to work out the degree four deformation) and show that the solutions to the Maurer Cartan are in fact the solutions for the deformation equations.
For the Lie algebras and modules over associative algebras cases, I am able to find the differentials defined on the Chevalley Eilenberg complex, $L_n = Hom_k(\wedge^{n+1}g, g)$ and the Hochschild complex for modules, $L_n = Hom_k(M\otimes B^{\otimes n}, M)$. However, I am unable to find the appropriate brackets for these two complexes.
For the Chevalley Eilenberg complex case, someone told me I should use the Schouten bracket. I have looked through a lot of papers and textbooks, but I cannot find the explicit formula of the bracket for my complex $L_n = Hom_k(\wedge^{n+1}g, g)$. When I search for Schouten bracket online, the materials are either too abstract for my problem or to difficult for me to understand.
I also encounter the same problem for the Hochschild complex for modules, $L_n = Hom_k(M\otimes B^{\otimes n}, M)$. On a paper by M. V. Ladoshkin, he defined the $\cup$ product on the complex $L_n = Hom_k(M\otimes B^{\otimes n}, M)$. The formula is $$f\cup g = (-1)^{(n+m-2)(m-1)}f(g\otimes \dots \otimes 1 \otimes 1),$$ $f$, $g$ in $L_n$ and $L_m$ respectively. However, when I use this product to mimic the Gerstenhaber bracket on associative algebra, i.e. $$[f,g] = f \cup g - (-1)^{nm}g \cup f,$$ the Maurer Cartan equation does not correspond to the deformation anymore. If $\mu: M\otimes A \to M$ sends $m \otimes a$ to $ma$, using the bracket above $[\mu, \mu]$ will only have one term and does not correspond to the condition $(a_1 \otimes a_0)m = a_1(a_0 m)$.
So my question is how to define the bracket (with explicit formulas) on $L_n = Hom_k(\wedge^{n+1}g, g)$ and $L_n = Hom_k(M\otimes B^{\otimes n}, M)$ to make them into DGLA's? Most of the papers and textbooks I have read define the differential for the two complexes, but they never talk about how to make the complex into a DGLA. If someone can give me the explicit formula for the brackets or point me to papers talks about how to define the brackets on these two complexes, it will be highly appreciated.