Israel-Gaza live updates: Israel 'deepening the war' in Gaza: Netanyahu

National Security Council spokesperson John Kirby called the mistaken fatal shootings of three Israeli hostages by the Israel Defense Forces "tragic,” adding there’s “no doubt” the Israelis will be looking into their rules of engagement.

“I have no doubt that they will do the forensics on this to learn what happened and how to avoid it happening again,” Kirby said. “It's tragic. It's sad. And you can't imagine that those IDF soldiers who pulled that trigger and killed those hostages feel very good about what they did. Of course not. It's a traumatic event.”

Kirby said sometimes the rules of engagement isn’t the issue, but rather “just the way they're enforced, or the interpretation of it by a unit on the ground or by an individual soldier.”

“That's why doing the forensics on this is going to be so important for them to kind of figure out, is there a systemic issue … or was this an individual issue? Misunderstanding, miscalculation, fog of war? I mean, we just don't know,” Kirby said. “I think we should be careful at this early stage, and certainly from here from Washington, to point the fingers at the exact rules of engagement.”

The IDF said it mistakenly killed three hostages who were carrying a stick with a white cloth during combat in Gaza, in what the head of the military said was "against the rules of engagement." The IDF said the three hostages were "mistakenly identified" as a threat.

The three men were identified as 28-year-old Yotam Haim; 26-year-old Alon Shamriz; and 22-year-old Samer Talalka.

-ABC News’ Justin Ryan Gomez

Israel Defense Forces spokesman Jonathan Conricus said Israel is closer today to war with Hezbollah, Lebanon's militant group, than yesterday, after a series of attacks over the past few days along Israel’s northern border.

"If we look at the amount of attacks and aggression that Hezbollah has mounted against Israel, more than a thousand different pieces of [ammunition] that have been fired at Israel, specifically by Hezbollah, then by any means, we could have been at war with Hezbollah long, long ago,” Conricus told reporters Monday. “And based solely on their actions, their violation of Israeli sovereignty and the casualties that they have caused … I think that we are -- and without being cheeky -- we are closer today to war than we were yesterday.”

Secretary of Defense Lloyd Austin said while in Israel he's offered "thoughts about how to transition from high intensity operations to a lower intensity and more surgical operations," and Austin said he and the Israelis have "had great discussions on all of those issues."

"In any operation like this, any campaign, there will be phases, and the most difficult part is as you shift from one phase to the next, making sure that you have everything accounted for and you get it right. So that requires detailed planning and very thoughtful planning," Austin said during a joint press conference in Israel Monday with Israeli Defense Minister Yoav Gallant.

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Defense secretary speaks out on Israel-Hamas war

ABC News’ Britt Clennett and Selina Wang break down the defense secretary’s comments on the Israel-Hamas war during a visit to Israel.

ABCNews.com

Gallant said their discussions have been "transparent and frank."

"On the personal level, I have a lot to learn from Gen. Austin," Gallant said. "He always gives me good advice."

Austin also reaffirmed America’s "unshakeable" support for Israel.

"At a time of mourning, a real friend shows up," Austin said. "And I know how terrible these days have been for the Israeli people. And I know that Oct. 7 touched everyone in this small democracy."

Austin emphasized the importance of limiting harm to civilians in Gaza, but he also stressed the "complexity" of the "battle space" in Gaza.

“Very closed spaces, dense population of people -- and so that makes it very, very difficult to conduct any military operation,” he said. "Above and beyond that, we see that Hamas routinely uses civilians as shields. Beyond that, they place their headquarters and their logistical sites near protected sites: hospitals, mosques, churches, you name it."

-ABC News’ Matt Seyler

Twenty-four aid trucks crossed through Israel's Kerem Shalom border crossing point into Gaza on Sunday, marking the first time aid had gone through this crossing since the war began, Palestinian Rafah border crossing spokesman Wael Abu Omar told ABC News.

Another 150 aid trucks crossed through the Egypt-Gaza Rafah border crossing on Sunday, the spokesman said.

The Kerem Shalom border crossing point closed after Hamas’ Oct. 7 attack on Israel. Last week, Israeli officials announced Kerem Shalom would open for security checks of aid trucks, but the trucks would still have to enter Gaza through the Rafah crossing. Then on Friday, Israel and the U.S. announced aid trucks would also be allowed to cross into Gaza from the Kerem Shalom crossing point.

-ABC News’ Ayat Al-Tawy