Classification of Certain Regular Subalgebras of sl (n, C) up to Conjugacy

In this paper we will be classifying some regular upper-triangular subalgebras of sl (n, C) up to conjugacy by matrices in SL (n, C). We do so for dimension 2, codimension 1, and codimension 2 subalgebras. We prove some general results for codimension k. The approach we use reduces an abstract classification problem to a combinatorial one, which we solve through a mixture of inductive and computational approaches.