Equality of the algebraic and geometric ranks of Cartan subalgebras and applications to linearization of a system of ordinary differential equations

If $L$ is a semisimple Lie algebra of vector fields on R^N with a split Cartan subalgebra C, then it is proved that the dimension of the generic orbit of C coincides with the dimension of C. As a consequence one obtains a local canonical form of L in terms of exponentials of coordinate functions and vector fields that are independent of these coordinates -- for a suitable choice of coordinates. This result is used to classify semisimple algebras of vector fields on R^3 and to determine all representations of sl(N, R) as vector fields on R^N. These representations are used to find linearizing coordinates for any second order ordinary differential equation that admits sl(3, R) as its symmetry algebra and for a system of two second order ordinary differential equations that admits sl(4, R) as its symmetry algebra.