In the classical theory of semisimple Lie algebras over the complex numbers (and elsewhere in Lie theory), it's convenient to apply easy Zariski density arguments for some underlying affine spaces. For instance, a natural proof of Harish-Chandra's basic theorem on the structure and characters of the center of the universal enveloping algebra involves restriction of polynomial functions from the Lie algebra to a Cartan subalgebra. Here the density of "regular" elements makes it possible to focus just on their behavior. Similarly, some classical conjugacy theorems for the Lie algebras relative to the adjoint group action are most easily studied in geometric terms. The point is that polynomials play a prominent role, making even the most elementary parts of algebraic geometry helpful.