Updated at 10:46 a.m. ET

The U.S. Supreme Court, in a 5-4 opinion, says the sedative used in Oklahoma's lethal injection cocktail does not violate the U.S. Constitution's ban on cruel and unusual punishment.

Here's the background to the case, in the words of SCOTUSblog:

"As part of its lethal injection protocol for executions, Oklahoma uses a sedative called midazolam, which is used to treat anxiety, to make the inmate unconscious. Two other drugs then paralyze him and stop his heart. A group of Oklahoma death-row inmates have challenged the state's use of midazolam, arguing that it cannot reliably render an inmate unconscious."The difference between midazolam and other drugs (like pentobarbitol) is that it's not a barbiturate. So the inmates' argument was that it wouldn't cause the sort of deep coma that would prevent pain later in the execution procedure."

The court ruled Monday that the death row inmates failed to establish that the use of midazolam violates the Eighth Amendment's ban on cruel and unusual punishment. The court's four liberal justices dissented.

Justice Sonia Sotomayor, in her dissent, criticized the court's majority opinion, saying it "leaves petitioners exposed to what may well be the chemical equivalent of being burned at the stake."

Justice Samuel Alito, writing for the majority, said the court affirmed a lower court's ruling for two reasons:

"First, the prisoners failed to identify a known and available alternative method of execution that entails a lesser risk of pain, a requirement of all Eighth Amendment method-of execution claims. ... Second, the District Court did not commit clear error when it found that the prisoners failed to establish that Oklahoma's use of a massive dose of midazolam in its execution protocol entails a substantial risk of severe pain."

And, he responded to Sotomayor's criticism, calling it "groundless."

"That is simply not true, and the principal dissent's resort to this outlandish rhetoric reveals the weakness of its legal arguments," Alito wrote.

As NPR's Nina Totenberg reported in April, it's the second time in seven years the court has looked at lethal injection — and it comes after several botched executions over the past year. Here's more from her story:

"Oklahoma was the first state to execute prisoners by lethal injection. The three-drug cocktail it developed and used for the first time in 1977 was soon adopted by every other capital punishment state. It was seen as the most humane way to impose the death penalty."The first drug in the original cocktail was sodium thiopental. It was used to put the prisoner into a deep, comalike state; the second was to paralyze him; and the third was to stop the heart."In 2008, the Supreme Court upheld that protocol. The court plurality said that while the second and third drugs were painful, the first drug, rendering the prisoner deeply unconscious, would prevent any great risk of pain."

But since that opinion, the drug used to put the prisoner into an insensate state has become difficult to obtain. Groups opposed to the death penalty have pressured manufacturers into refusing to provide sodium thiopental or a similar drug called pentobarbital for use in executions. So some death penalty states have turned to midazolam, which hasn't been approved by the FDA, to put people into a deep, coma-like state. That was the drug used in the botched 2014 execution of Clayton Lockett in Oklahoma. That case prompted the state to change its execution protocol; it now uses a larger dose of midazolam, along with a paralytic agent and potassium chloride.

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