Commutator in Nilpotent group of class 2 is "bilinear"

Question

I found the following statement about the "bilinearity" of the commutator in a nilpotent group of class 2 (ie. $[G,G']=1$, where $G'=[G,G]$) in a paper, but I do not see how it is true.

Let $G$ be a (finite) nil-2 group, then for every $g_1,g_2,g_3,g_4\in G$, $$[g_1g_2,g_3g_4]=[g_1,g_3][g_1,g_4][g_2,g_3][g_2,g_4].$$

It is claimed that this easily follows from the fact that $G'\leq Z(G)$. Is there something that I am missing, and is this even true?

Answer 1

You could use the identity $[xy, z] = y^{-1}[x,z]y [y,z]$ and the similar identity for $[x, yz]$.