I have a doubt on Freudenthal Triangulation. I want to partition a simplex into finer simplices.

The FT gives me the vertices of the simplices which partition my original simplex into finer simplices.

I am interested in enumerating over each of the simplices obtained from the FT algorithm.

However, I am not able to conclude that given the vertices, how do I construct the dividing simplices out of them. In other words, if the vertices are given to me then how can I find which of these vertices make one of the simplices (that partitioned the original simplex)

I believe it might be a simple question with simple solution but I am not able to figure it out. Can you please point me to some reference. Thank You