Let $A$ be an associative algebra in $dgVect_k$. Then the commutator $[\cdot,\cdot]:A\otimes A\to A$ defined by $[x,y]=xy-(-1)^{|x||y|}yx$ gives $A$ the structure of a (dg-)Lie algebra. The **Maurer--Cartan equation**:
$$d\alpha+\frac 12[\alpha,\alpha]=0$$
for $\alpha\in A$ (necessarily of degree $1$) plays an important role in deformation theory.

Does anyone recognize the following generalization of the above construction from associative algebras (that is, $E_1$-algebras) to $E_n$-algebras?

Let $A$ be an $E_n$-algebra in $dgVect_k$. The $E_n$-algebra structure gives a map $A\otimes A\otimes C_\bullet(Conf_2(D^n))\to A$ where $Conf_2(D^n)\simeq S^{n-1}$ is the configuration space of two distinct points in $D^n$. By picking a cycle in $C_\bullet(Conf_2(D^n))$ representing $[S^{n-1}]$, we get a pairing $[\cdot,\cdot]:A\otimes A\to A$ which I will think of as a sort of "higher commutator" (of course, this recovers the usual notion of commutator when $n=1$).

Now for some questions:

Does this "higher commutator" endow $A$ with the structure of a dg-Lie algebra (or, more likely, an $L_\infty$-algebra), or is it something more exotic?

What is the significance of the solutions of the Maurer--Cartan equation (using the higher commutator) to the given $E_n$-algebra?