SPRINGFIELD — Halfway through arguments to uphold Drew Peterson’s conviction for killing his third wife, Justice Mary Jane Theis wanted clarity.

Prosecutors had alleged Peterson, the former Bolingbrook police officer who made headlines when his fourth wife went missing in 2007, was motivated to kill his previous wife because of a messy split.

He wanted to avoid divvying up his property and splitting time with the couple’s children. And he could do that by preventing her from testifying during the divorce case, state lawyers had claimed.

But Theis’ questioning on that point during oral arguments in the case today suggested some members of the Illinois Supreme Court may not be convinced.

“So, could you tell us, what is the evidence that you are directing us to, to show not just generally a motive to kill, but specifically, an intent to silence the witness?” Theis asked.

Assistant Attorney General Leah Myers Bendik replied that Peterson became agitated when he learned his wife, Kathleen Savio, might get their marital house and half their marital property.

Bendik noted Savio was found dead only about a month before their divorce trial was scheduled to begin, in April 2004. And she noted that Peterson received custody of the children and got his uncle to stand in as administrator of her estate after her death.

“Are you arguing that if she had appeared to testify, the result of the divorce case would’ve been different?” Theis asked.

Bendik said that was “definitely true.” She ultimately agreed that it was circumstantial evidence, but that it was still relevant. She said the trial judge in the case, Will County Associate Judge Edward A. Burmila Jr. was in the best position to determine whether Peterson’s intent was clear, and that the Supreme Court shouldn’t alter the decision because it was not an abuse of discretion.

“What we have here is — defendant completely thwarted the judicial process of the divorce case with Kathy by getting her out of the system and preventing her from continuing to be present to object to his approach to the settlement, to his desire about the custody of the children,” Bendik said.

Among other things, Peterson’s attorneys, Harold J. Krent, dean of the IIT Chicago-Kent College of Law, and Steven A. Greenberg, of Steven A. Greenberg Ltd., argued that Peterson’s trial was enormously flawed.

They argued that statements from divorce attorney Harry C. Smith, in which he claimed Peterson’s fourth wife, Stacy, said she knew her husband had killed his previous wife, should not have been allowed due to attorney-client privilege.

“Harry Smith violated the rules on attorney-client privilege,” Greenberg told the justices. “Attorney-client privilege is one of those sacred things that an attorney should never violate.”

He mentioned that a media contract between Peterson and one of his attorneys, Joel A. Brodsky, who represented him between 2007 and 2012, was a clear conflict of interest.

The deal allowed them to collect fees along with a media firm that would help them get interviews and book and movie deals as Peterson’s case became more notorious.

Savio’s death was initially ruled accidental after she was found in a bathtub. But the case was later ruled a homicide, with Peterson being charged in 2009. He was ultimately found guilty in September 2012 and sentenced to 38 years in prison, although there was a lack of physical evidence linking him to the murder.

A few years into his sentence, Peterson was charged with a murder-for-hire plot aimed at Will County State’s Attorney James W. Glasgow, who had prosecuted him in the Savio case. He was convicted of that crime earlier this year and sentenced to another 40 years.

No charges were ever filed in Stacy’s 2007 disappearance.

The case is People v. Drew Peterson, No. 120331.