  For every associative algebra $A$ and every class $\mathcal{C}$ of
representations of $A$ the following question (related to nullstellensatz)
makes sense: Characterize all tuples of elements $a_1,\ldots,a_n \in A$ such
that vectors $\pi(a_1)v,\ldots,\pi(a_n)v$ are linearly dependent for every $\pi
\in \mathcal{C}$ and every $v$ from the representation space of $\pi$. We
answer this question in the following cases:
  (1) $A=U(L)$ is the enveloping algebra of a finite-dimensional complex Lie
algebra $L$ and $\mathbb{C}$ is the class of all finite-dimensional
representations of $A$.
  (2) $A=U(\mathfrak{sl}_2(\mathbb{C}))$ and $\mathbb{C}$ is the class of all
finite-dimensional irreducible representations of $A$.
  (3) $A=U(\mathfrak{sl}_3(\mathbb{C}))$ and $\mathbb{C}$ is the class of all
finite-dimensional irreducible representations of $A$ with sufficiently high
weights.
  In case (1) the answer is: tuples that are linearly dependent over
$\mathbb{C}$ while in cases (2) and (3) the answer is: tuples that are linearly
dependent over the center of $A$. Similar results have been proved before for
free algebras and Weyl algebras.
