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Realizing the difficulty of grasping what is meant in Romans 8:26 as the Interceding Work of the Holy Spirit, I gathered together some of the comments I found on the same at biblos.com.
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King James Bible
Likewise the Spirit also helpeth our infirmities: for we know not what we should pray for as we ought: but the Spirit itself maketh intercession for us with groanings which cannot be uttered.
Maketh intercession - The word used here ὑπερεντυνγχάνει huperentungchanei, occurs no where else in the New Testament. The word ἐντυνγχάνω entungchanō, however, is used several times. It means properly to be present with anyone for the purpose of aiding, as an advocate does in a court of justice; hence, to intercede for anyone, or to aid or assist in any manner. In this place it simply means that the Holy Spirit greatly assists or aids us; not by praying for us, but in our prayers and infirmities.
...but the Spirit itself maketh intercession, for us, with groanings which cannot be uttered; not the spirit of a man; or the gift of the Spirit in man; or a man endued with an extraordinary gift of the Spirit; but the Holy Ghost himself, who makes intercession for the saints: not in such sense as Christ does; for he intercedes not with the Father, but with them, with their spirits; not in heaven, but in their hearts; and not for sinners, but for saints: nor in the manner as Christ does, not by vocal prayer, as he when on earth; nor by being the medium, or way of access to God; nor by presenting the prayers of saints, and the blood and sacrifice of Christ to God, as Christ does in heaven; nor as the saints make intercession for one another, and for other persons: but he intercedes for them, by making them to intercede; he indites their prayers for them, not in a book, but in their hearts; he shows them their need, what their wants are; he stirs them up to prayer, he supplies them with arguments, puts words into their mouths, enlarges their hearts, gives strength of faith in prayer, and all the ardour and fervency of it; he enables them to come to God as their Father; and gives them liberty and boldness in his presence, which requires an heart sprinkled from an evil conscience, faith in the blood and righteousness of Christ, and a view of God, as a God of peace, grace, and mercy: and this intercession he makes, "with groanings which, cannot be uttered"; not that the Spirit of God groans, but he stirs up groans in the saints; which suppose a burden on them, and their sense of it: and these are said to be "unutterable"; saints, under his influence, praying silently, without a voice, as Moses and Hannah did, 1 Samuel 1:13, and yet most ardently and fervently; or as not being able to express fully what they conceive in their minds, how great their burdens are, and their sense of their wants.
Geneva Study Bible:
(h) Incites us to pray, and tells us as it were within, what we will say, and how we will speak.
PEOPLE'S NEW TESTAMENT:
But the Spirit itself. The Spirit himself (Revised Version). Observe the climax: The creation groans; we ourselves groan; the Spirit himself groans. The Spirit within us intercedes by groaning which are his, in that they are prompted by the Spirit. Augustine says:
It is not in himself, nor in the substance of the Eternal and Blessed Trinity that he groans, but in us because he makes us groan.''
Groanings which cannot be uttered. Speechless groanings.Wesley's Notes:
We know not - Many times. What we should pray for - Much less are we able to pray for it as we ought: but the Spirit maketh intercession for us - In our hearts, even as Christ does in heaven. With groanings - The matter of which is from ourselves, but the Spirit forms them; and they are frequently inexpressible, even by the faithful themselves.
But the Spirit himself intercedes,  etc. Though really or by the event it does not appear that our prayers have been heard by God, yet Paul concludes, that the presence of the celestial favor does already shine forth in the desire for prayer; for no one can of himself give birth to devout and godly aspirations. The unbelieving do indeed blab out their prayers, but they only trifle with God; for there is in them nothing sincere, or serious, or rightly formed. Hence the manner of praying aright must be suggested by the Spirit: and he calls those groanings unutterable, into which we break forth by the impulse of the Spirit, for this reason -- because they far exceed the capability of our own minds.  And the Spirit is said to intercede, not because he really humbles himself to pray or to groan, but because he stirs up in our hearts those desires which we ought to entertain; and he also affects our hearts in such a way that these desires by their fervency penetrate into heaven itself. And Paul has thus spoken, that he might more significantly ascribe the whole to the grace of the Spirit. We are indeed bidden to knock; but no one can of himself premeditate even one syllable, except God by the secret impulse of his Spirit knocks at our door, and thus opens for himself our hearts.
EDITOR to JOHN CALVIN'S COMMENTARY:
"Intercedit -- huperentunchanei -- abundantly intercedes," for so huper, prefixed to verbs, is commonly rendered. This is the proper action of an advocate, a name given to the Spirit by our Savior, allon parakleton -- "another advocate," not "comforter," as in our version, and Christ is called by the same name in 1 John 2:1, and the same work, "interceding," is ascribed to him, Hebrews 7:25. But we learn in John 14:16, that the Spirit is an advocate with us -- "that he may abide with you for ever;" and in 1 John 2:1, that Christ is an advocate in heaven -- "with the Father." The same name and a similar kind of work are ascribed to both. Some, as Doddridge, to avoid the blending the offices of the two, have rendered the verb here by a different term, but not wisely. -- Ed.