"Blood Atonement Doctrine"
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Bruce R. McConkie, Mormon Doctrine, pp. 92-93.
... the true doctrine of blood atonement is simply this:
1. Jesus Christ worked out the infinite and eternal atonement by the shedding of his own blood. He came into the world for the purpose of dying on the cross for the sins of the world. By virtue of that atoning sacrifice immortality came as a free gift to all men, and all who would believe and obey his laws would in addition be cleansed from sin through his blood. (Mosi 3:16-19; 3 Ne 27:19-21; 1 Jn 1:7; Rev 5:9-10.)
2. But under certain circumstances there are some serious sins for which the cleansing of Christ does not operate, and the law of God is that men must then have their own blood shed to atone for their sins. Murder, for instance, is one of these sins; hence we find the Lord commanding capital punishment. Thus, also, if a person has so progressed in righteousness that his calling and election has been made sure, if he has come to that position where he knows "by revelation and the spirit of prophecy, through the power of the Holy Priesthood" that he is sealed up unto eternal life (D&C 131:5), then if he gains forgiveness for certain grievous sins, he must "be destroyed in the flesh," and "delivered unto the buffetings of Satan unto the day of redemption, saith the Lord God." (D&C 132:19-27.)
President Joseph Fielding Smith has written: "Man may commit certain grievous sins -- according to his light and knowledge [italics in original] -- that will place him beyond the reach of the atoning blood of Christ. If then he would be saved, he must make sacrifice of his own life to atone -- so far as in his power lies -- for that sin, for the blood of Christ alone under certain circumstances will not avail.... Joseph Smith taught that there were certain sins so grievous that man may commit, that they will place the transgressor beyond the power of the atonement of Christ. If these offenses are committed, then the blood of Christ will not cleanse them from their sins even though they repent. Therefore their only hope is to have their own blood shed to atone, as far as possible, in their behalf." (Doctrines of Salvation 1:133-138.)
This doctrine can only be practiced in its fulness in a day when the civil and ecclesiastical laws are administered in the same hands. It was, for instance, practiced in the days of Moses, but it was not and could not be practiced in this dispensation, except that persons who understood its provisions could and did use their influence to get a form of capital punishment written into the laws of the various states of the union so that the blood of murderers could be shed.