After signing the Magna Carta, was the king of England able to execute a noble and his family and escheat their property to the crown by his own will up to 1688? For example in the cases such as Lèse-majesté, was the king able to kill a high rank noble (e.g. a duke) by his direct order? Or it needed to run a judicial procedure in a court? If there were any cases such as I said, could you introduce me some of them?
OK, I've done some research, and I can't find any cases of high-ranking nobles and their families being executed by the monarch after Magna Carta.
In fact, there were surprisingly few extra-judicial executions after 1215. The exceptions seem to have been mostly during the Wars of the Roses (1455 - 1487). Richard Neville, 16th Earl of Warwick was allegedly executed following the Battle of Barnet (although this might be construed as being part of the battle), but his family were not targeted for revenge, despite his acts of rebellion.
A more egregious case was that of William Hastings, 1st Baron Hastings, who was summarily executed without even a pretence of a trial by Richard, Duke of Gloucester (the future Richard III) in 1483. But even in this case, Richard made no effort to persecute Hastings' widow and children. In fact, Richard explicitly allowed Lady Hastings to retain her husband’s land and goods, with the sole exception of Hastings’ royal grants, which were seized and returned to the crown.
Apart from these few exceptions, English monarchs seem to have been pretty scrupulous in their application of the law in these cases. Even Henry VIII had to follow a form of procedure (often through acts of attainder passed by Parliament). Of course, as @TheHonRose has pointed out in the comments above, Parliament was likely to do what it was told in the case of a powerful king like Henry VIII. Nevertheless, he still had to go through the process of the law to get what he wanted.
Oxford History of England:
- Volume IV: The Thirteenth Century, 1216–1307 — Sir Maurice Powicke
- Volume V: The Fourteenth Century, 1307–1399 — May McKisack
- Volume VI: The Fifteenth Century, 1399–1485 — E. F. Jacob
- Volume VII: The Earlier Tudors, 1485–1558 — J. D. Mackie
- Volume VIII: The Reign of Elizabeth I, 1558–1603 — J. B. Black
- Volume IX: The Early Stuarts, 1603–1660 — Godfrey Davies
- Volume X: The Later Stuarts, 1660–1714 — Sir George Clark
Most of the time the King couldn't just order "off with his head!" but instead had to have a sham trial and conviction before killing a noble. The execution of nobles after often sham trials was so common that at the end of the reign of Elizabeth I there weren't any dukes left in England. They had all been beheaded and their titles forfeited.
There are a number of exceptions to the rule of nobles being tried and convicted before being killed by the king.
Eleanor, sister and heiress of Arthur of Brittany, and rightful Queen of England from 1203 to 1241, died in captivity in 1241 and some suspected she had been killed without a trial by order of Henry III.
Deposed king Edward II was reported to have died of natural causes in prison at Berkeley Castle but many believed he had been killed without a trial.
Deposed King Richard II was reported to have died of natural causes in prison, but many believed he had been killed without a trial.
17-year-old Edmund, Earl of Rutland, was either killed in battle in 1460 or else captured and put to death by the Lancastrians, probably without a trial.
17-year-old Edward, Prince of Wales, was either killed in battle in 1471 or else captured and put to death by the Yorkists, probably without a trial.
Deposed King Henry VI died in the Tower of London in 1471 soon after Prince Edward, allegedly of grief, though modern examination of his skeleton showed signs of violence - no doubt ordered by Edward IV without a trial.
The last legitimate male member of the Beaufort family was captured by the Yorkists in battle in 1471 and put to death a few days later, probably without a trial.
12-year-old King Edward V was deposed in 1483 and, along with his 9-year-old brother Richard, disappeared from sight in the Tower of London, being presumed to have been killed without a trial.
Henry VIII's vengeance on the Pole family included sending little Henry Pole to the Tower where he vanished from the record, being presumed to have been starved to death without a trial.
The bones of two children were discovered in the Tower in the 17th century and presumed to be those of Edward V and Richard Duke of York. A few decades earlier Sir Walter Raleigh recorded the discovery of two children's skeletons in the Tower when he was imprisoned, so perhaps there were two "foul and midnight murders" - ordered by kings - of pairs of children in the Tower.
This was before 1215, but it illustrated the power of the king. In this instance, it was King Henry II who reportedly said, "Will no one rid me of this meddlesome priest?. This was a reference to Thomas a Becket, Archbishop of Canterbury. Some four knights took this complaint as a "command" and assassinated the Archbishop. Basically, there was always someone in kingdom willing to do the king's dirty work, as evidenced by a number of mysterious deaths over the centuries, especially of young children or deposed rulers with potential rival claims to the throne, as MAGolding pointed out.
So the king (or queen) basically had the ability to have anyone they wanted killed, even if it were technically not an "execution." The main reason why a king might go through a formal trial and execution of someone is because of the further punishment of "attainder" that is the right to disinherit the condemned man's family and seize his estates, because of "corruption of blood."