Commutative subalgebras of $U(\mathfrak{q})$ of maximal transcendence degree

We prove that the enveloping algebra $U(\mathfrak{q})$ of a finite-dimensional Lie algebra $\mathfrak{q}$ contains a commutative subalgebra of the maximal possible transcendence degree $(\dim \mathfrak{q}+\mathrm{i}\mathrm{n}\mathrm{d}\mathfrak{q})\mathrm{/}2$.