"For two centuries after the Conquest, the frank, open rebellions of the great folk were treated with a clemency which, when we look back to it through the intervening ages of blood, seem wonderful."
The structure and function of civil conflict is a subject of some fascination to me, particularly in the context in which we find ourselves in Crusader Kings II and III. The above quote and the example of Waltheof, Earl of Northumbria stand for England, but that's not to say that the Norman conquerors were an entirely forgiving breed. The conspiracy to overthrow William Rufus and crown Stephen of Aumale gives us this memorable sentence for one of the conspirators: "the king ordered William of Eu who had been defeated in a duel to have his eyes put out and to be castrated, and his steward, William of Aldrie, the son of his aunt, and privy to his treason, to be hanged." And it's not just CK2 players who deliberately let people languish in their dungeons until death, King John did it, too: "The Annals of Margam state that William de Braose the younger, with his wife, several sons, and Matilda his mother, were captured by John in Ireland, and first imprisoned at Bristol, and afterwards at Windsor. Fifty thousand marks were fixed as the price of their redemption. William the father being allowed his liberty in order to obtain the ransom, fled to France, and thereupon the king starved his wife and son to death."
Jourdain de l'Isle-Jourdain, lord of Casaubon, married to a niece of the French pope John XXII, was hanged for rape, rapine, and murder in 1323, but the exceptional nature of the act was noted even in his own time. "One cannot remember nor can one find it written in the gestes of France that a man as high-born as my lord Jourdain has ever suffered such a death since the time of Ganelon." Execution was, and should be, a rare punishment for the high and low nobility. But I'm not saying to let them go with a slap on the wrist and a warning not to get caught next time: executions were rare, because actions deemed sufficiently heinous as to deserve it were also rare. Exile, fines, imprisonment, and confiscation were all customary means to deal with political rivals that sought to orchestrate one's removal from the throne: this is a cost of doing business. Uneasy lies the head that wears a crown, and all that. Pointedly, conspiracies to revolt are rather more likely than conspiracies to commit infanticide.
Still, a vein of ruthlessness can be found in rulers of the age, even when it came those with rank equal to their own. Charles of Anjou, that fabulous adventurer and conqueror of Sicily, had Conradin, his 16-year old rival for the Sicilian crown (nephew to king Manfred, dead in battle against Charles' army), and also duke of Swabia and king of Jerusalem, beheaded alongside his close companion Frederick, Margrave of Baden. Count Giordano Lancia, two counts Gherardesca, and several other close advisors and supporters died with them. Pope Clement IV is said to have signed off on the sentence, to boot: "The life of Conradin is the death of Charles; the death of Conradin is the life of Charles." If a ruler is sufficiently ruthless and his grasp on the throne sufficiently perilous, then he may take extraordinary measures to permanently end the claims of his rivals. See also: Richard III and his nephews. King Ramiro II of Aragon presents us with a similar example: "When they had arrived in Huesca, the king ordered certain of his confidential servants to arm themselves in the chamber, and he told them what they were to do there. When the nobles and knights had arrived, he ordered them to be called to council individually, one after the other. When they entered this room, he ordered those inside to be beheaded immediately. Only those who were guilty in their actions toward him were so called. In this fashion, twelve knights and nobles were beheaded before the king dined. In truth, he would have beheaded all the other nobles and knights, except that those who were outside somehow found out what was happening, and fled. When these were dead, the others who had fled were powerless, and Ramiro's realm remained quiet, and in security and peace."
In short, a great deal should depend on the context of the offense, the laws of the realm, and the king's character. Yes, you should be able to freely execute those who've taken the blood of the royal family in an act of naked murder, but such crimes should also be extremely rare. In terms of more conventional conflict, a realm with a high degree of baronial independence and a tradition of autonomy is unlikely to countenance the killing of your political opponents even after a rebellion. A king fresh to the throne, crown unsteady on his head, might get away with fearful acts of brutality, cowing the nobility into obedience for some years to come, but may face consequences later in his reign for his harsh actions. Ramiro II of Aragon gave up his realm in favor of his son-in-law to retreat to monastic life shortly after cementing control of the realm. Charles of Anjou was expelled from Sicily by an uprising, the infamous Sicilian Vespers. King John faced the Barons' War and the Magna Carta and ultimately died on campaign.