I just did this stuff in class and realized that there's an easy enough proof of Cauchy-Binet, assuming the most basic properties of determinants, that it shouldn't be considered an obstruction. Compute the determinant of the matrix [Id_n B, A 0_k] where A is kxn and B is nxk. By row and column reduction, it's (-1)^k det(AB). By direct expansion, of the (n+k)! terms, only (n choose k)*k!^2 of them are not obviously zero, and those sum to (-1)^{k^2} times the RHS of Cauchy-Binet.

It's here arxiv.org/abs/math/0306275

Stefan, it's easy to prove that any connected smooth manifold is acted on transitively by its diffeomorphism group. (Connect two points by a path, trivialize in a tubular nbhd of the path, ... ) duetosymmetry, I'm more worried about your statement "finite-dim (real) Lie algebras are classified". If you mean semisimple ones, then yeah, but to give an idea of the horrors, let R act on V=R^2 by speed 1 rotation, and on W=R^2 by speed sqrt(2) rotation, then make the semidirect product R x (V+W). Irrational-flow-on-the-torus issues means that this has coadjoint orbits that aren't locally closed.

Prakash Belkale, Invariant theory of 𝐺 𝐿 (𝑛) and intersection theory of Grassman- nians. Int. Math. Res. Not. (2004), no. 69, 3709–3721.

Qiaochu's right assuming $End(V)$ means the monoid under multiplication, admittedly not the first meaning one might guess from the notation.