Let $\mathfrak{g}$ be an $L_\infty$-algebra (over a characteristic zero field $k$). then, if $\mathfrak{g}$ is finite-dimensional in each degree, the $L_\infty$-algebra $inn(\mathfrak{g})$ can be defined as the $L_\infty$-algebra whose Chevalley-Eilenberg algebra is the Weil algebra $W(\mathfrak{g})$ of $\mathfrak{g}$. The Weil algebra has a remarkable freeness property:

$Hom_{dgca}(W(\mathfrak{g}),\Omega^\bullet)=Hom_{dgVect}(\mathfrak{g}^*[-1],\Omega^\bullet)=\Omega^1(\mathfrak{g}),$where $\Omega^\bullet$ is an arbitrary dgca and $\Omega^i(\mathfrak{g})$ denotes the vector space of degree $i$ elements of $\Omega^\bullet\otimes\mathfrak{g}$. The identification can then be used to define *curvature* of $\mathfrak{g}$-connections, and to write down their Bianchi identities. Indeed

and the underlying graded vector space of $inn(\mathfrak{g})$ is $\mathfrak{g}\oplus\mathfrak{g}[1]$, so that the identification together with the freeness property of $W(\mathfrak{g})$ gives the following:

for any $A\in \Omega^1(\mathfrak{g})$ there exists a unique $F_A\in \Omega^2(\mathfrak{g})$ such that the pair $(A,F_A)$ satisfies the Maurer-Cartan equation in $\Omega^\bullet(inn(\mathfrak{g})$. The element $F_A$ is the curvature of $A$ and the Maurer-Cartan equation expresses both the relation between $A$ and $F_A$ and the Bianchi identities.

To write these equations in a fully explicit form in a way that allows making contact with classical equations from differential geometry, it would be convenient to have the $L_\infty$-algebra structure on $inn(\mathfrak{g})$ spelled out in terms of lots of brackets (in terms of the lots of brackets defining the $L_\infty$-algebra sturcuture on $\mathfrak{g}$). Do we have these brackets already spelled out somewhere?

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