The split mind, the part of the One Mind which has lost itself in the thought of separation, projects the world and loses itself in that projection, similarly to how we can daydream a scenario and lose ourselves within our mind into that daydream without ever leaving our outer reality.
The Course speaks of this as a personalized process, one that can be interpreted as being initiated by the separate self, but the separate self in the dream is not what dreams. Nor is it separate from the dreamer. The self loses itself in its own thoughts and seems to have an existence all its own. But this happens within the mind thinking it, in much the same way we can lose ourselves in our own daydream by getting caught up in a drama of our own imagining. The me in which the daydream arises and the me in the dream are not separate, and yet it is the me in the daydream who must recognize it’s in a dream it no longer wants, and return itself back to its self outside the dream. The return is facilitated not by the self which identifies as being separate, but by the self that is both in the dream and outside of it. They are one and the same, but are not experienced as such by the self which thinks that the dream is its reality. Because of this, the Course speaks to the self on two levels: it speaks to the separate self because that is the level where help is needed; but also to the encompassing Self because that is where return is found.
“Can God’s Son lose himself in dreams, when God has placed within him the glad Call to waken and be glad? He cannot separate himself from what is in him. His sleep will not withstand the Call to wake. The mission of redemption will be fulfilled as surely as the creation will remain unchanged throughout eternity. You do not have to know that Heaven is yours to make it so. It is so. Yet to know it, the Will of God must be accepted as your will.” (T-13.XII.10)