Too tired to think clearly, but it looks like a standard Deformation Theory thingy.

We have natural linear maps $\gamma_n (L)\rightarrow gr_n L$. Split them as linear maps. We get a bijective linear map $L\rightarrow gr L$.

Use it to equip $gr L$ with a second Lie algebra structure $[,]^{\prime}$, coming from $L$. Now consider the difference $$\mu:L\otimes L \rightarrow L, \ \ \mu (x\otimes y) = [x,y]-[x,y]^{\prime}$$ I claim that $\mu$ is a 2-cocycle on $gr L$ and that finding an isomorphism $\gamma_n (L)\cong gr_n L$ is equivalent to finding a 1-cochain $\theta$ such that $\mu=d\theta$.

$\DeclareMathOperator\gr{gr}$Too tired to think clearly, but it looks like a standard Deformation Theory thingy.

We have natural linear maps $\gamma_n (L)\rightarrow \gr_n L$. Split them as linear maps. We get a bijective linear map $L\rightarrow \gr L$.

Use it to equip $\gr L$ with a second Lie algebra structure $[,]^{\prime}$, coming from $L$. Now consider the difference $$\mu:L\otimes L \rightarrow L, \ \ \mu (x\otimes y) = [x,y]-[x,y]^{\prime}.$$ I claim that $\mu$ is a 2-cocycle on $\gr L$ and that finding an isomorphism $\gamma_n (L)\cong \gr_n L$ is equivalent to finding a 1-cochain $\theta$ such that $\mu=d\theta$.