He is on trial for the alleged murders of Sacramento sheriff's deputy Danny Oliver and Placer County sheriff's detective Michael Davis Jr. during a 2014 crime spree.
White denied Barbour's request and said the trial would indeed continue.
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The chilling outburst led Sacramento Superior Court Judge Steve White to briefly remove the jury from the court room and then warn Bracamontes against any further interjections.
White previously ruled that Bracamontes was mentally competent, but Barbour asked for the judge to reconsider his decision.
His wife, Janelle Monroy, is also charged in the slayings of Oliver and Placer County Sheriff's Deputy Michael Davis Jr. during a daylong crime spree.
Bracamontes, 37, is from Mexico and is in the United States illegally.
Barbour and Dawson then renewed their efforts, asking the judge if they could enter a plea of not guilty by reason of insanity.
Before the verbal threats by the illegal alien were made, Bracamontes' defense attorney told the jury that the Mexican national was responsible for the deaths of Davis and Oliver.
He is on trial along with his wife, Janelle Monroy, 41.
He admits killing two police officers, injuring another and shooting a bystander in the head in an attempt to steal his auto. One point that his legal team will argue is that a blood test after his arrest showed that Bracamontes had methamphetamine in his system.
"Forgive me, God", the note read. "Please take me with you". President Trump has weighed in on the case, mentioning it during a speech to Congress previous year and introducing the widow of one of the slain deputies. Instead, Bracamontes crawled out of the house and surrendered meekly.
Bracamontes has a long history of acting out during court hearings, and did so several times Tuesday on the first day of trial.
"There was no need to prove all this s--", he said in court.