  The universal enveloping algebra $\mathscr{U}$ of a two-dimensional
nonabelian Lie algebra $L$ is a Lie algebra itself with the commutator as Lie
bracket. There exists a presentation of $\mathscr{U}$ with generators $x,y$ and
relation $xy-yx=x$ such that the Lie subalgebra of $\mathscr{U}$ generated by
$x,y$ is isomorphic to $L$, which is only a two-dimensional vector subspace of
the infinite-dimensional $\mathscr{U}$. Much then of the Lie structure of
$\mathscr{U}$ is ubiquitous, yet unexamined when the characteristic of the
scalar field is zero. In such a case, we show that there exists a linear
complement of $L$ in $\mathscr{U}$ that contains an infinite-dimensional Lie
subalgebra of $\mathscr{U}$ for which we give a presentation by generators and
relations. We extend this Lie subalgebra into a filtration of $\mathscr{U}$.
