= Solution
Write the <lower central series of a Lie algebra> as $\mathfrak g^1=\mathfrak g$ and $\mathfrak g^{j+1}=[\mathfrak g,\mathfrak g^j]$. Since the <nilpotent Lie algebra> is nonzero, there is a last nonzero term $\mathfrak g^c$, with $\mathfrak g^{c+1}=0$. Thus $[\mathfrak g,\mathfrak g^c]=0$, so
$$
0\ne\mathfrak g^c\subseteq Z(\mathfrak g).
$$
Consequently \b[the centre is nonzero]. This argument works over every <field> and includes an abelian algebra, for which $c=1$.
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