For my thesis, I'm trying to understand the Poisson operad (I'll call it $Pois$) and its homotopy counterpart $Pois_\infty$. 

In Loday-Valette, it is stated that $Pois=Comm\circ Lie$, the operad structure relying on the existence of a distributive law $\lambda:Comm\circ Lie\rightarrow Lie\circ Comm$. Somewhere on the internet, I've read that $Pois_\infty$ can be defined in a similar way, as the composition $E_\infty \circ Lie$ (if I remember correctly). 

I was wondering if there's any reference for this (if it is correct). Also, can we defined $Pois_n:=E_n\circ Lie$? What if we use the cofibrant replacement $Lie_\infty$ (the $L_\infty$ operad) instead of $Lie$, do we still have an operad structure? I assume by homotopy theory of operads that using $Lie_\infty$ is not a problem, but I'm not so sure.