If (say) $L$ is an $L_\infty$-algebra, then it is known that under certain conditions there exists a quasi-isomorphic $L_\infty$-algebra $L'$ which is a differential graded Lie algebra: all ternary and higher brackets of $L'$ vanish. This is sometimes called *rectification*.

(I am a physicist, so I assume $L$ is over a field of characteristic $0$; quasi-isomorphisms are then morphisms whose linear parts induce isomorphisms on cohomology. I wouldn't mind hearing about $L$ over formal power series rings, however.)

There exist lots of operad-powered general proofs, such as in this paper by Berger and Moerdijk. Unfortunately, those are not accessible to me, so I am looking for operad-free explicit constructions/algorithms producing $L'$ and the associated quasi-isomorphism. I am also interested in the case where $L$ is endowed with a cyclic inner product, should that make a difference.

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