WASHINGTON—The Supreme Court approved Trump administration plans to execute a second inmate this week, rejecting by a 5-4 vote claims that dementia had left Wesley Ira Purkey unable to comprehend the reason for his punishment, therefore rendering it unconstitutional.

In a one-sentence order issued after 2 a.m. on Thursday, the court’s conservative majority lifted lower court orders temporarily blocking the execution so that Mr. Purkey, 68 years old, could pursue those claims. The court’s four liberal members dissented.

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