Past exam of the Lie algebras course of the University of Oxford / 2011 / 1 / i / 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 $[\mathfrak{g},{\mathfrak{g}}^c]=0$, soConsequently the centre is nonzero. This argument works over every field and includes an abelian algebra, for which $c=1$.