Embedding tensors on Lie ∞-algebras with respect to Lie ∞-actions

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Given two Lie ∞ -algebras E and V, any Lie ∞ -action of E on V defines a Lie ∞ -algebra structure on E ⊕ V . Some compatibility between the action and the Lie ∞ -structure on V is needed to obtain a particular Loday ∞ -algebra, the non-abelian hemisemidirect product. These are the coherent actions. For coherent actions it is possible to define non-abelian homotopy embedding tensors as Maurer-Cartan elements of a convenient Lie ∞ -algebra. Generalizing the classical case, we see that a non-abelian homotopy embedding tensor defines a Loday ∞ -structure on V and is a morphism between this new Loday ∞ -algebra and E.