Potere would not say whether the decision to seek a death sentence against McAndrew suggested that investigators doubted his claims of mental illness.
His attorney Stephen Heckman described their decision as "ill-advised."
"Given the facts in this case and the mental-health issues, I don't know if he fully understands it," he said. "We hope we can convince prosecutors not to seek the death penalty."
Since his arrest last spring, McAndrew has been held at Norristown State Hospital, a facility used to house criminal suspects in need of mental health treatment.
Although prosecutors have argued that they have no record of his ever having received a professional diagnosis before his incarceration, a judge has denied their requests that McAndrew be returned to a prison cell.
Family friends maintain he has long struggled with what they believe to be schizophrenia.
Upper Merion police found McAndrew on March 5 covered in blood and standing outside his family's house in the Gulph Mills section of the township. Inside, his father, Joseph Sr.; mother, Susan; and twin brother, James, lay stabbed and bleeding alongside what detectives have described as an 18-inch "samurai-style" sword.
During initial interviews with investigators, McAndrew called the deaths an "extermination" of a "person named brother," "person named mother," and "person named father," according to court documents.
McAndrew is set to take his case before a jury Oct. 4.
Contact staff writer Jeremy Roebuck at 267-564-5218, jroebuck@phillynews.com, or @jeremyrroebuck on Twitter. Read his blog, "MontCo Memo," at www.philly.com/montcomemo 
