This compound of 5 components (1 octahedron + 4 tetrahedra) can well be considered connected back into a single component, if each pair of coplanar faces rather becomes considered as a single compound entity instead.

Incidence matrix

4  * * |  3  0  0 |  3 0 0  (hidden)
* 12 * |  1  2  0 |  2 1 0
*  * 6 |  0  0  4 |  0 2 2
-------+----------+-------
1  1 0 | 12  *  * |  2 0 0
0  2 0 |  * 12  * |  1 1 0
0  0 2 |  *  * 12 |  0 1 1
-------+----------+-------
1  2 0 |  2  1  0 | 12 * *
0  3 3 |  0  3  3 |  * 4 *  2{3}-compound
0  0 3 |  0  0  3 |  * * 4