I have noted that some people do not believe in the doctrine of trinity, namely "three personas, one nature", but my question is not about whether the doctrine is true!
This question is also not about what "of the same nature, but different personas" actually means. That question was asked by me at What does it mean that Jesus, Father, and Holy Spirit are of the same nature but different personas? However, if you feel that answering, what it means, is required for answering this question, I am happy about clarification.
This question is also not about how to argue for the trinity, I asked this at How is the doctrine of the Trinity substantiated?
This question is about, how how those, that believe the doctrine of trinitarianism explain the following verses. Thus this is not a question of opinion.
When I do the search
I actually do not get phrases in which Jesus is sitting at the right hand of the father, but instead the right hand side of God, or the power of God and similar, e.g. Acts 7, 56.
How does this coincide with the doctrine of the trinity? According to the doctrine of the trinity I understand that Jesus and the Father are personas of God. While I cannot claim to understand what that means, I would intuitively expected that Jesus should be sitting next to the Father.
https://www.gotquestions.org/right-hand-God.html mentions that
Therefore, what we can say is that "God’s right hand" refers to the Messiah, the LORD Jesus Christ and He is of equal position, honor, power and authority with God (John 1:1-5).
Also here I would have expected to see "Father" instead of "God".
On the other hand it seems that Jesus "went" to the father
And Jesus, crying out with a loud voice, said, "Father, into your hands I commit my spirit." Having said this, He breathed His last. (Luke 23:46)
and by with the Father
For just as the Father has life in Himself, even so He gave to the Son also to have life in Himself. (John 5:26)
The question is why the texts say "God's right hand" rather than "the Father's right hand". Is "God’s right hand" a title / expression but not to be understood as a description of the relationship of God/Jesus/Father?
(This questions was inspired by https://hermeneutics.stackexchange.com/q/47522, but is different.)