  The complete reducibility property for bipartite states reduced the
separability problem to a proper subset of positive under partial transpose
states and was used to prove several theorems inside and outside entanglement
theory. So far only three types of bipartite states were proved to possess this
property. In this work, we provide some procedures to create states with this
property, we call these states by the name of completely reducible states. The
convex combination of such states is the first procedure, showing that the set
of completely reducible states is a convex cone. We also provide a complete
description of the extreme rays of this set. Then we show that powers, roots
and partial traces of completely reducible states result in states of the same
type. Finally, we consider a shuffle of states that preserves this property.
This shuffle allows us to construct states with the complete reducibility
property avoiding the only three conditions known to date that imply this
property.
