While this passage may mention the Father, the Son, and the Holy Spirit, it does not, however, teach that these are divisions within the Godhead. We find no declaration in this verse that the Father is true God, the Holy Spirit is true God, and the Son is true God, and that all three exist as co-eternal, co-equal members of the same being. If Matthew 28:19 is left as it is, what we have is the instruction of Jesus to baptize in the authority of God, Jesus, and the Holy Spirit.
The Jewish crowds in Jerusalem had recognized that Jesus was the king of Israel who came “in the name of Yahweh” (Matt 21:9), that is, that he was coming to them with God’s authority. Similarly, about the angel who wielded God’s authority in Exodus 23:21, God said “my name is in him.” The phrase “in the name of” actually appears several times in Matthew as a reference to a given subject’s authority: “Many will say to me on that day, Lord, Lord, did we not prophecy in your name, and in Your name cast out demons, and in Your name perform many miracles?” (Matt 7:22). The authority into which one is baptized need not be God; Paul states that the Israelites “were all baptized into Moses” (1 Cor 10:2), or, by the authority that Moses’ name carries. Jesus’ prescription to baptize in the authority of both the Father and the Son, and also in the authority carried by the Holy Spirit, is similar to a police officer commanding someone to stop “in the name of the Law,” or by the authority that the Law carries. There is no clear and necessary Trinity here. (Kegan A. Chandler, The God of Jesus in Light of Christian Dogma: The Recovery of New Testament Theology [McDonough, Ga.: Restoration Fellowship, 2016], 343)