I am looking for examples of prime ideals for Lie algebras. In particular, I am interested in examples involving the Lie algebra given by the commutator of endomorphisms of a complex vector space and the Lie algebra of functions on the phase space $\mathbb{R}^{2n}=(x^1,\ldots,x^n,p_1,\ldots,p_n)$ given by $$ [f,g]=\sum_i\frac{\partial f}{\partial x^i}\frac{\partial g}{\partial p_i}- \frac{\partial g}{\partial x^i}\frac{\partial f}{\partial p_i}, $$ where $f$ and $g$ are real or complex-valued functions on the phase space.
My interest has originated in physics, where the time derivative of a physical "observable" $X$ can generally be written (except for multiplicative constants) as $$ \frac{d X}{d t} = [H,X], $$ where $H$ is the Hamiltonian operator in quantum mechanics or the Hamiltonian function on classical phase space. If the Hamiltonian and $X$ are not in the prime ideal at some initial time, then $X$ will never be.
I have made some searching on the internet, but I have found only two papers that are rather abstract with no examples at all.