Classification of Certain Regular Subalgebras of $\mathfrak{s}\mathfrak{l}(n,\mathbb{C})$ up to Conjugacy

In this paper we will be classifying some regular upper-triangular subalgebras of $\mathfrak{s}\mathfrak{l}(n,\mathbb{C})$ up to conjugacy by matrices in $SL(n,\mathbb{C})$. 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.