B-RNS-GSS formalism and $L_{\mathrm{\infty }}$-actions

Pure spinor formalism and RNS formalism are related by a chain of equivalences constructed by introducing and integrating-out BRST quartets. This is known as B-RNS-GSS formalism. One of the steps can be understood as adding auxiliary fields to lift a strong homotopy action of the SUSY Lie superalgebra in the large Hilbert space to a strict action. We develop a general prescription for this ``strictification’’ procedure, which can be applied for any strong homotopy action of a Lie superalgebra. We explain how it is related to the B-RNS-GSS formalism.