for $\dim{\mathfrak z }({\mathfrak n }_{m,k})$ we construct elements of length $k$ in a Hall basis $\mathcal B$.

Let $\{e_1,\ldots,e_m\}$ a set of generators of ${\mathfrak n }_{m,k}$ and consider the set $$\begin{aligned}
\mathcal U &=&\{[[[e_i,e_j],e_k],e_m]: 1\leq j<i\leq m,\,k\geq j\}.\nonumber  $$

Any element in $\mathcal U$ is basic and of length 4. Given $x\in \mathcal U$, the bracket $$[x,e_m]^{(s)}:= [[[x,\overbrace{e_m],e_m]\cdots , e_m]}^{s}  \;\;s\geq 1$$ is an element in the Hall basis if $\ell([x,e_m]^{(s)})\leq k$.

In fact if $s=1$ then

-   both $x=[[[e_i,e_j],e_k],e_m] \in \mathcal U$ and $e_m$ are elements of the Hall basis, and $x >e_m$ because of their length;

-   also $x=[G,H]$ with $G=[[e_i,e_j],e_k]$ and $H=e_m$ and we have $e_m\geq H$.

So both conditions of the Hall basis definition are satisfied and hence $[x,e_m]^{(1)}\in \mathcal B$ and it belongs to $ C^4({\mathfrak n }_{m,k})$.

Inductively suppose $[x,e_m]^{(s-1)}\in\mathcal B$, then clearly $[[[x,e_m],e_m]\cdots ], e_m]^{(s-1)}>e_m$ and it is possible to write $[x,e_m]^{(s-1)}=[G,H]$ with $H=e_m$. Thus $[x,e_m]^{(s)}\in\mathcal B$. Notice that $[x,e_m]^{(s)}\in C^{s+3}({\mathfrak n }_{m,k})$.

We construct the new set $$\widetilde{\mathcal U}:=\{[x,e_m]^{(k-4)}:x\in \mathcal U\}\subseteq C^{k-1}({\mathfrak n }_{m,k}),$$ which is contained in the center of ${\mathfrak n }_{m,k}$ and it is linearly independent. Therefore $$\label{u}
\dim {\mathfrak z }({\mathfrak n }_{m,k}) \geq |\widetilde{\mathcal U}|.$$

