God is one in His essence, and an important ramification of this is that whenever He acts in relation to things outside Himself (ad extra), each of the three persons acts inseparably with the others. This manner of working is not akin to the work of a committee, wherein different members each contribute their different gifts and talents to a cooperative endeavor, for the oneness of God means that the attributes of the three persons are identical. Father, Son, and Holy Spirit all have the same will, wisdom, power, holiness, and so forth. Yet, this manner of inseparable operations is also not a monochrome activity wherein you cannot distinguish the three persons of the Trinity. Each person exercises the same divine attributes, but each does so in a manner that is peculiar to His person.
This is difficult to conceptualize, but it is somewhat easier to conceive when we look at the work of atonement. Although it is the incarnate Son, touching His human nature, who atones for our sin, all three persons work inseparably to effect the atonement that secures our salvation. Both Father and Son offer up the Son for our redemption, the Father as the subject who offers and the Son as both the subject who offers and the object who, touching His human nature, is offered (Rom. 8:31–32; Heb. 9:13–14). And when the Father and Son offer up the Son, They do so in the Spirit, who by His "efficacious power" makes Christ's death as a man under divine wrath saving for us (Calvin; see Heb. 9:13–14). Atonement is from the Father through the Son who is offered in the Spirit for our salvation. It is a work of holy love by all three persons of the Trinity.
Yet, the inseparable work of the divine persons does not exclude uniqueness among Them. One early church heresy, patripassianism, collapsed the particularity of each Trinitarian person's manner of acting in the inseparable operation of atonement by saying that both Father and Son suffered. This was wrong because the particularity of the Son's manner of acting and His unique possession of a human nature means only Christ suffered. However, Christ did not suffer with respect to His deity. The divine nature cannot change, so it cannot be subjected to suffering. Christ suffered on the cross only touching His human nature, not His divine nature. Human nature is changeable, so it can be subjected to suffering, and the Father did not unite to His person a human nature but only the Son. So, the Father did not suffer on the cross but only Christ, and then only touching His humanity.
Touching His divine nature, it was impossible for Christ to die. Touching His human nature, Christ could and did die. Christ could atone for our sin because He was truly man, but He was offered up for our salvation by all three members of the Trinity. In the humanity of Christ, God bore His own curse against sin, and now we live because of His work. Let us praise our Creator for His salvation.