A Lie ring is a triple $(G,+, [\ ,\ ]),$ where $(G,+)$ is an abelian group and $ [\ ,\ ]$ is a bilinear map satisfying

• $[x,x]=0$
• $[\ ,\ ]$ is bilinear
• $[[x,y],z]+[[y,z],x]+[[z.x],y]=0,\ \forall\ x,y,z\in G.$ (Jacobi identity).

The center of the Lie ring is defined as $$Z(G)=\{x\ |\ [x,y]=0,\ \forall\ y\in G\}.$$

Let $(G,+, [\ ,\ ])$ be a Lie ring with free presentation $$0\rightarrow R\rightarrow F\rightarrow G\rightarrow 1.$$ Now consider the quotient Lie ring $\tilde{F}=\frac{F}{R\cap [F,F]}$ and $\tilde{R}=\frac{R}{R\cap [F,F]}$. Then the subring $Z(\tilde{F})\cap [\tilde{F},\tilde{F}]$ of $Z(\tilde{F})$ has a complement subring in $Z(\tilde{F})$, i.e. there exists an ideal $\tilde{T}$ of $Z(\tilde{F})$ such that $$Z(\tilde{F})=Z(\tilde{F})\cap [\tilde{F},\tilde{F}]\times \tilde{T}.$$

I have seen in some article that authors have taken as assumption that if $\tilde{x}=x+[R,F]\in\frac{F}{[R,F]} $ such that $\tilde{x}^n\in \frac{[F,F]}{[R,F]}$ for some integer $n\in \mathbb{N}$ then $\tilde{x}\in \frac{[F,F]}{[R,F]}$ and assumed that this holds. But I am not able to conclude it. Please guide me in this regard.