Eric Williams was sentenced to death in December after his estranged wife testified how she and Williams murdered Kaufman County District Attorney Mike McLelland, Mike's wife, Cynthia, and prosecutor Mark Hasse.
At Wednesday's court hearing, a doctor testified on Williams' behalf, saying brain scans showed Williams had a prior brain trauma and abnormalities that affected his emotional state, moral character, and decision making.
Williams' attorneys say that's evidence that could have affected the outcome of the trial.
They also say those abnormalities could have been caused by diabetes.
Williams was in the courtroom briefly before asking to be excused from the hearing. He went back to his cell.
The hearing continues Thursday.