It all depends on your understanding of mans Dichotomy. I have never found it necessary to believe Jesus died in the Spirit, or that His Spirit died. In all the Theology I have poured through, it seems the flesh dies and the spirit of the righteous goes to be with Him who gave it. Jesus rose again because He Is The Life, He is The Creator of Life, and The Author of Life. As God: It is impossible for Him to die, and He alone possesses immortality, so we perceive that from our observations: that He left His body at death and that his body remained dead, until He indwelt his tent again, i.e. His resurrected His body."Jesus died on the cross, right? Did God? Thank you for any and all responses" (Brin)
When a believer dies, he is dead, his body is dead, and for all practical semantics we remain dead even though we are alive with Christ in the Spiritual realm. The death on the cross was the death of the flesh (symbolic of our own death to the flesh, and not unlike circumcision). Without Christ/God no one can be resurrected to 'life'. Only believers will see life again. Although the dead 'may' be 'conscious', believers and unbelievers are not 'among' the living, and are indeed dead (until the believer receives their immortal body).
All three are One, and indeed all three indwell in Christ. But the substance and or nature of God are not the same as the persons, who are unique in themselves, and are distinct individuals who can talk to each other (One God, three persons). The incarnation seems to be a time when Jesus as a person was separate from the persons of the Holy Spirit and the Father. God does have person-hood, yet what person-hood 'looks like or consists of' can only be defined and understood as accurately and empirically as we can see and understand what spirit is. We can affirm that God and men have both person-hood and spirit, but what persons and spirits actually 'look like and consists of' is still beyond our ability to see or understand (I suppose if we could see or comprehend 'spirit' it may alter our definition of spirit). Just as: Gods fullness is beyond our ability to see our see or understand, yet we do know some things about God. And we can be sure of what God has 'revealed' about both the body and spirit from His revealed Word .'The Son became a human body, but whose spirit is the Holy Spirit?'
The incarnation seems to also be a time when Jesus was separate from His own Glory, and a time when Jesus had to rely on the on the other two persons of God for Divine knowledge and possibly Divine power, just as we do. Still, because of the power of a sinless life, death could not hold the Author of Life.