But if ye have respect to persons, ye commit sin - You transgress the plain law of God, and do wrong. See the references on James 2:1.
And are convinced of the law as transgressors - Greek "By the law." The word convinced is now used in a somewhat different sense from what it was formerly. It now commonly refers to the impression made on a man's mind by showing him the truth of a thing which before was doubted, or in respect to which the evidence was not clear. A man who doubted the truth of a report or a proposition may be convinced or satisfied of its truth; a man who has done wrong, though he supposed he was doing what was proper, may be convinced of his error. So a man may be convinced that he is a sinner, though before he had no belief of it, and no concern about it; and this may produce in his mind the feeling which is technically known as conviction, producing deep distress and anguish. See the notes at John 16:8. Here, however, the word does not refer so much to the effect produced on the mind itself, as to the fact that the law would hold such an one to be guilty; that is, the law pronounces what is done to be wrong. Whether they would be personally convinced of it, and troubled about it as convicted sinners, would be a different question, and one to which the apostle does not refer; for his object is not to show that they would be troubled about it, but to show that the law of God condemned this course, and would hold them to be guilty. The argument here is not from the personal distress which this course would produce in their own minds, but from the fact that the law of God condemned it.

But if ye have respect to persons - In judgment, or in any other way; ye commit sin against God, and against your brethren, and are convinced, ελεγχομενοι, and are convicted, by the law; by this royal law, Thou shalt love thy neighbor as thyself; as transgressors, having shown this sinful acceptance of persons, which has led you to refuse justice to the poor man, and uphold the rich in his oppressive conduct.

But if ye have respect to persons, ye commit sin, &c. This is not doing well, but is a transgression of the law, as every sin is; hence it follows,
and are convinced of the law as transgressors; which carries on a formal process against such persons; it accuses them of sin, and charges them with it; it proves it upon them, and convicts them of it; it pronounces them guilty, and curses them for it; and passes the sentence of condemnation and death upon them; wherefore care should be taken not to commit this sin, and so fall under the convictions and reproofs of the law.

Respect of persons violates the command to love all alike "as thyself."
ye commit sin--literally, "ye work sin," Matthew 7:23, to which the reference here is probably, as in James 1:22. Your works are sin, whatever boast of the law ye make in words (see on James 2:8).
convinced--Old English for "convicted."
as transgressors--not merely of this or that particular command, but of the whole absolutely.

Being convicted - By that very law. Exodus 23:3.