The world has its ideas of what “divisive” means. It assigns blame for “divisiveness” based on its ideas.
The preface to the Book of Concord identifies corrupting the purity of the heavenly doctrine as the cause of division. Satan
scattered the seeds of false doctrine and dissensions in the churches and schools [Matthew 13:24-40]. He also labored to stir up divisions [Romans 16:17] combined with offense. By these arts of his, he labored to corrupt the purity of the heavenly doctrine, to sever the bond of Christian love and godly agreement [Ephesians 4:3].
The editors who prepared the Reader’s Edition of the Book of Concord just quoted inserted the Scripture references in brackets. Note their reference to Romans 16:17. There, the Apostle Paul, and God himself, given the office of apostle and the doctrine of Scripture, have ideas of what “divisive” means and who causes division.
Paul says, “Now I urge you, brethren, note those who cause divisions and offenses, contrary to the doctrine which you learned, and avoid them.” (Romans 16.17)
In Paul’s idea, division is a matter of doctrine. Candy coated exteriors, diplomatic manners, and the like are not unifying. The one catholic – universal – faith is unifying. Division of doctrine is divisive.
Look at Lenski’s translation of that verse.
Now I admonish you, brethren, to look out for those causing the divisions and the death-traps contrary to the doctrine which you on your part did learn and definitely incline away from them.
The divisions are definite. They are “the” divisions. They are “the” death-traps. They are death-traps leading to death. There is little new or novel in them. They are the same old, same old errors and heresies. They cycle back again around the world and through the centuries.
The substantivized present participle [means] “engaged in the business of producing,” and what they make it their business to produce is “standing apart,” divisions, the opposite of unity or “thinking the same thing in accord with Christ,” and “with one accord with one mouth glorifying God” (15:5, 6).
Worse than that, they also cause … “death-traps”; see the word in 9:3 and 11:9, and note that it always designates what is fatal. “Offenses” (A.V.) can be understood only in the sense of mortal offenses; and “occasions of stumbling (A.V.) must be discarded. These errorists tear believers out of the unity of the church, and their teachings often act like deathtraps in which souls are fatally caught.
As Lenki says, the word “doctrine” is in disrepute in some circles. But, he goes on,
Note well that the apostolic doctrine never causes either inward or outward rents in the church, either division of mind or schism in communion and fellowship. How can it when it is ever one and the same? Being one, it unifies, holds unity. When those who hold this doctrine firmly reject those who refuse to hold it or some part of it, they cause no division but prevent division by not giving room to those who do divide and disunite. They keep the unity in the Word intact against those who would invade and disrupt that unity.
As to that, Pieper say,
Such, however, as separate from a church body because it tenaciously clings to false doctrine are unjustly called schismatics, separatists, etc. This separation is commanded in Scripture (Rom. 16:17) and is the only means of restoring and maintaining the true unity of the Christian Church.
Kretzmann comments on the same verse,
It is not the open enemies of the Christian Church that work the greatest harm, but the false teachers that call themselves after the name of Christ and purport to believe in, and to teach, the Bible, and who, by insidious propaganda, subvert the foundations of sound teaching.
Both Lenski and Kretzmann exposit Paul’s words as admonishing us to keep our eyes open, to keep a look out for the false teachers, then to mark them and decisively avoid and have nothing to do with them. The admonition is not to sift their false teachings, cherry pick some good things from them, and use those in teaching our children, in our devotional books, in our sermons, in our Sunday School classes, in our day school curricula, in our synodical catechisms, or the publications of our synodical publishing house. No, we are to look out, mark, and avoid. This is for the sake of unity and life.
Walther carries the admonition to the following extent:
A man may proclaim the pure doctrine, but if he does not condemn and refute the opposing false doctrine, does not warn against the wolves in sheep’s clothing, the false prophets, and unmask them, he is not a faithful steward of God’s mysteries, not a faithful shepherd of the sheep entrusted to him, not a faithful watchman on the walls of Zion, but, as the Word of God says, an unfaithful servant, a dumb dog, a traitor.
If we think that Paul, Lenski, Kretzmann, Pieper, and Walther are just cranky old men, consider that the “apostle of love” goes further than any of them, saying, “If anyone comes to you and does not bring this doctrine, do not receive him into your house nor greet him; for he who greets him shares in his evil deeds.” (2 John 10-11) As Tyndale translates John, the apostle of love says, “nether bid him God spede.”
What is the loving thing to do? Pieper says
To say that love demands [fellowship with heterodoxy] is a misuse of that word. Love of God and love of the brethren rather requires the opposite practice. He who loves Christ loves Christ’s Word, and Christ command us to avoid all who teach anything that is contrary to His Word. And whoever really loves the brethren refuses to participate in their erring and sinning, seeking rather to deliver them from error and sin.
 “Preface to the Christian Book of Concord,” Concordia: The Lutheran Confessions, 2nd ed. (St. Louis: Concordia Publishing House, 2006), 3 (Scripture references in brackets in the CPH Reader’s Edition as set forth in the quotation here).
 Richard C. H. Lenski, The Interpretation of St. Paul’s Epistle to the Romans (Minneapolis: Augsburg Publishing House, 1936), 913-914.
 Lenski, 914.
 Lenski, 916-917.
 Francis Pieper, Christian Dogmatics, vol. III (St. Louis: Concordia Publishing House, 1950), 427
 Paul E. Kretzmann, Popular Commentary of the Bible, New Testament, II (St. Louis: Concordia Publishing House, 1924), 83.
 Carl Ferdinand Wilhelm Walther, Pastorale, 82 as quoted in Pieper, op cit. 50.
 Pieper, vol. III, 425.