I don’t know if it could be said that God separated part of himself. My understanding of this issue (which is limited, at best) is that Jesus is a different person from God the Father, but of the same substance. Likewise for the Holy Spirit, so that there are three persons, but only one substance.

In much the same manner, Jesus himself is one person with two natures: one human and one Divine. “…concerning his Son Jesus Christ our Lord, which was made of the seed of David according to the flesh (human nature); And declared to be the Son of God with power (Divine nature), according to the spirit of holiness, by the resurrection from the dead….” (Romans 1.3-4).