A strong constructive version of engel's theorem

We examine two problems in the computational theory of Lie algebras. First, we prove a constructive version of Engel's theorem: if L is a finite-dimensional Lie algebra that is not nilpotent, we show how to construct an element x in L such that the linear transformation ad x is not nilpotent. No special assumptions about the underlying field are needed. Second, as an important application of the first result, we give an algorithm for the construction of a Cartan subalgebra of a finite-dimensional Lie algebra. This solves the problem of finding a totally constructive proof of the existence of a Cartan subalgebra, posed by Beck, Kolman, and Stewart in the paper "Computing the Structure of a Lie Algebra". Our proofs are ordinary mathematical proofs that do not employ the general law of excluded middle. The advantage of this approach to mathematics is that our proofs, which are not burdened or obscured by the details of a particular programming language, can nevertheless be routinely turned into computer programs