For my thesis, I'm trying to put the structure of an $L_\infty$-algebra on a tensor product $A\otimes L$ of a $L_\infty$-algebra $L$ with a differential, graded commutative algebra $A$. I know that there are arguments by abstract homotopical nonsense to prove the existence of such a structure, but I'm looking for a more explicit approach.
My idea is to define the higher brackets as $$\tilde l_n(\alpha_1\otimes v_1,\cdots,\alpha_n\otimes v_n)=\eta \alpha_1\cdots\alpha_n l_n(v_1,\cdots,v_n)$$ while the $1$-brackets are $\tilde l_1(\alpha\otimes v)=d\alpha\otimes v+(-1)^{|\alpha|}\alpha\otimes l_1(v)$. Here $\eta=\eta(\alpha_1\otimes v_1,\cdots,\alpha_n\otimes v_n)=(-1)^{\sum_{1\leq i<j\leq n}|v_i||\alpha_j|}$, so that, for example, $\eta(\alpha_1\otimes v_1,\alpha_2\otimes v_2)=(-1)^{|v_1||\alpha_2|}$.
My question is if this is something already in the literature (I know it is the way brackets and differentials are defined when $L$ is a dg Lie algebra).
