Question about triangles and vectors,Question about getting triangle pairs from a mesh

Hello, currently i'm using unity to develop a game in c#... Basically i have 2 triangles with random positions in a 3d space... These triangles have and index and array of their vertices positions, so...

 public class Triangle
 {
     public int index;
     public Vector3[] _transform = new Vector3[3];
 }

when i create the 2 triangle objects i would like to check if they are neighbours, meaning if they have 2 vertices in commun, then i would like to save their indexes and the 2 coincident points so...

 public class TriPairs
 {
     public Vector2Int pair;
     public Vector3[] points = new Vector3[2];
 }

i tried making a nested for loop for comparison but it just increases in complexity when trying to compare and extract the points, any ideas? Thanks in advance