Lower-central commutators add weights

All ${\gamma }_i$ are characteristic: an automorphism preserving ${\gamma }_i$ preserves the generating commutators for $[G,{\gamma }_i]$; start at $G$. Also $[U,V]=[V,U]$ because $[u,v{]}^{-1}=[v,u]$. For $i=1$, $[G,{\gamma }_j]={\gamma }_{j+1}$ is the required inclusion.

Induct on $i$, uniformly for all $j\ge 1$. Normality and the three-subgroup containment give $[{\gamma }_{i+1},{\gamma }_j]\le [{\gamma }_{j+1},{\gamma }_i][[{\gamma }_j,{\gamma }_i],G]$. By symmetry and the induction hypothesis these two factors lie respectively in ${\gamma }_{i+j+1}$ and $[{\gamma }_{i+j},G]={\gamma }_{i+j+1}$. This proves the inclusion for $i+1$ and every $j$.

Each ${\gamma }_i\mathrm{/}{\gamma }_{i+1}$ is central in $G\mathrm{/}{\gamma }_{i+1}$ and therefore abelian. Replacing $x\in {\gamma }_i$ by $xu$ with $u\in {\gamma }_{i+1}$ changes $[x,y]$ only by commutators of weight at least $i+j+1$ and conjugations of $[x,y]$. The latter also change it only by $[G,{\gamma }_{i+j}]\le {\gamma }_{i+j+1}$. The same argument replaces $y$ by $yv$, $v\in {\gamma }_{j+1}$. Thus the displayed map is independent of representatives.

Modulo ${\gamma }_{i+j+1}$ the conjugations in both product identities disappear, so the pairing sends a product in either input to the product of its values. Identity inputs give identity output; inverses give inverse outputs by applying the product rule to $x{x}^{-1}$. This is biadditivity for the abelian factors, for all positive indices, also when any factor is trivial.