Journal of New Theory, no.43, pp.83-91, 2023 (Peer-Reviewed Journal)
By operationalizing Fn as a free Lie Algebra of finite rank n, this work considers the orbit problem for Fn. The orbit problem is the following: given an element u ∈ Fn and a finitely generated subalgebra H of Fn, does H meet the orbit of u under the automorphism group AutFn of Fn? It is proven that the orbit problem is decidable for finite rank n, n ⩾ 2. Furthermore, we solve a particular instance of the problem – i.e., whether H contains a primitive element of Fn. In addition, some applications are provided. Finally, the paper inquires the need for further research.