Transferring human obedience, creaturely obedience, into the life of God implies his creaturehood. That implication must be rejected. As the Bible tells us and consent of the church has confirmed, the Father and Son are distinguished by Fatherness and Sonness. Their relation is one of Fatherness and Sonness.
In 2016 Evangelicals debated about the best way to affirm that God is one and yet Father and Son. The old answer is: the Father begets the Son eternally; the Son is eternally begotten. Beget and begotten are old words to describe how fathers generate children. A mother births them; a father begets.
In recent years, evangelicals attempted to find a new way to talk about Father and Son. They said that the Father relates to the Son because he has paternal authority; the Son relates to the Father in a mode of submission. Authority and submission distinguish Father and Son.
For the most part, people found the new approach insufficient. It implied eternal inferiority of the Son, implied two wills, and inserted the human life of Jesus where he obeyed the Father into God. It unintentionally implied a creaturely characteristic in God since Jesus’s creaturely obedience to the Father gets imported into how God is eternally!
Recently, however, a theologian reaffirmed that the Father eternally has authority over the eternally submissive Son. Interestingly, the theologian cited Augustine and Hilary of Poitiers as proponents of his position.
Two Reasons Why Eternal Submission Does Not Work
First, Jesus submits to the Father in his role of Mediator, one who became obedient to the point of death in the form of a slave (Phil 2:7). But he was equal to the Father in the form of deity (Phil 2:6).
To transfer submission into God as the way the Father and Son differ is to transfer a creaturely characteristic into God. Because Jesus took on humanity, he obeys the Father vicariously in his role of Mediator for our sake.
Second, the church Fathers such as Augustine and Hilary made the above distinction clearly. They affirmed the obedience of the Son according to his humanity. But they did not pass through this obedience into God to explain how the Son and Father eternally related.
Just one example. Augustine in The Trinity writes: “In the form of a servant which he took he is the Father’s inferior; in the form of God in which he existed even before he took this other he is the Father’s equal.” Elsewhere, he says “the Father is greater than is the form of the servant, whereas the Son is his equal in the form of God.”