Find shared properties of a cluster samples

I have a dataset which contains ~15 features. With the elbow method, I found out that the optimal number of clusters is probably four. Therefore, I applied the K-means algorithm with four clusters. Now, I would like to understand why these clusters have been formed the way they are. In other words, I would like to know what are the shared properties of the points of this cluster.

My idea is the following:

Let’s pretend that C1 are the coordinates of the centroid of the first cluster and that P1 and P2 are two points of this cluster.

$ $ C1 = \begin{pmatrix} 5\ 2\ 4\ \end{pmatrix} $ $

$ $ P1 = \begin{pmatrix} 8\ 2\ 6\ \end{pmatrix} P2 = \begin{pmatrix} 9\ 2\ 0\ \end{pmatrix} $ $

If we compute the average distance of the different coordinates of P1 and P2 we obtain this:

$ $ DistAverage = \begin{pmatrix} ((8-5)+(9-5))/2\ ((2-2)+(2-2))/2\ ((6-4)+(4-0))/2\ \end{pmatrix} = \begin{pmatrix} 3.5\ 0\ 3\ \end{pmatrix} $ $

Would this mean that the second feature is a “shared property” of the points of this cluster (since the average distance is 0) ?

I hope that the question was clear enough.