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

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