How to realize a Lie algebra by vector fields

Abstract

We describe an algorithm for embedding a finite-dimensional Lie algebra (superalgebra) into a Lie algebra (superalgebra) of vector fields that is suitable for a ground field of any characteristic and also a way to select the Cartan, complete, and partial prolongations of the Lie algebra of vector fields using differential equations. We illustrate the algorithm with the example of Cartan’s interpretation of the exceptional simple Lie algebra $$\mathfrak{g}$$ (2) as the Lie algebra preserving a certain nonintegrable distribution and also several other examples.