A Commentary, Critical, Practical, and Explanatory on the Old and New Testaments, by Robert Jamieson, A.R. Fausset and David Brown  at sacred-texts.com
deu 21:1EXPIATION OF UNCERTAIN MURDER. (Deu 21:1-9)
If one be found slain . . . lying in the field, and it be not known who hath slain him--The ceremonies here ordained to be observed on the discovery of a slaughtered corpse show the ideas of sanctity which the Mosaic law sought to associate with human blood, the horror which murder inspired, as well as the fears that were felt lest God should avenge it on the country at large, and the pollution which the land was supposed to contract from the effusion of innocent, unexpiated blood. According to Jewish writers, the Sanhedrin, taking charge of such a case, sent a deputation to examine the neighborhood. They reported to the nearest town to the spot where the body was found. An order was then issued by their supreme authority to the elders or magistrates of that town, to provide the heifer at the civic expense and go through the appointed ceremonial. The engagement of the public authorities in the work of expiation, the purchase of the victim heifer, the conducting it to a "rough valley" which might be at a considerable distance, and which, as the original implies, was a wady, a perennial stream, in the waters of which the polluting blood would be wiped away from the land, and a desert withal, incapable of cultivation; the washing of the hands, which was an ancient act symbolical of innocence--the whole of the ceremonial was calculated to make a deep impression on the Jewish, as well as on the Oriental, mind generally; to stimulate the activity of the magistrates in the discharge of their official duties; to lead to the discovery of the criminal, and the repression of crime.
deu 21:10THE TREATMENT OF A CAPTIVE TAKEN TO WIFE. (Deu 21:10-23)
When thou goest to war . . . and seest among the captives a beautiful woman . . . that thou wouldest have her to thy wife--According to the war customs of all ancient nations, a female captive became the slave of the victor, who had the sole and unchallengeable control of right to her person. Moses improved this existing usage by special regulations on the subject. He enacted that, in the event that her master was captivated by her beauty and contemplated a marriage with her, a month should be allowed to elapse, during which her perturbed feelings might be calmed, her mind reconciled to her altered condition, and she might bewail the loss of her parents, now to her the same as dead. A month was the usual period of mourning with the Jews, and the circumstances mentioned here were the signs of grief--the shaving of the head, the allowing the nails to grow uncut, the putting off her gorgeous dress in which ladies, on the eve of being captured, arrayed themselves to be the more attractive to their captors. The delay was full of humanity and kindness to the female slave, as well as a prudential measure to try the strength of her master's affections. If his love should afterwards cool and he become indifferent to her person, he was not to lord it over her, neither to sell her in the slave market, nor retain her in a subordinate condition in his house; but she was to be free to go where her inclinations led her.
deu 21:15If a man have two wives, one beloved, and another hated--In the original and all other translations, the words are rendered "have had," referring to events that have already taken place; and that the "had" has, by some mistake, been omitted in our version, seems highly probable from the other verbs being in the past tense--"hers that was hated," not "hers that is hated"; evidently intimating that she (the first wife) was dead at the time referred to. Moses, therefore, does not here legislate upon the case of a man who has two wives at the same time, but on that of a man who has married twice in succession, the second wife after the decease of the first; and there was an obvious necessity for legislation in these circumstances; for the first wife, who was hated, was dead, and the second wife, the favorite, was alive; and with the feelings of a stepmother, she would urge her husband to make her own son the heir. This case has no bearing upon polygamy, which there is no evidence that the Mosaic code legalized.
deu 21:18If a man have a stubborn and rebellious son--A severe law was enacted in this case. But the consent of both parents was required as a prevention of any abuse of it; for it was reasonable to suppose that they would not both agree to a criminal information against their son except from absolute necessity, arising from his inveterate and hopeless wickedness; and, in that view, the law was wise and salutary, as such a person would be a pest and nuisance to society. The punishment was that to which blasphemers were doomed [Lev 24:23]; for parents are considered God's representatives and invested with a portion of his authority over their children.
deu 21:22if a man have committed a sin . . . and thou hang him on a tree--Hanging was not a Hebrew form of execution (gibbeting is meant), but the body was not to be left to rot or be a prey to ravenous birds; it was to be buried "that day," either because the stench in a hot climate would corrupt the air, or the spectacle of an exposed corpse bring ceremonial defilement on the land.