Israel continued to strike southern Lebanon on Tuesday as Iran-backed Hezbollah attacked its troops there despite an apparent Washington-brokered de-escalation deal and a fourth round of U.S.-hosted talks between Lebanon and Israel.

Prime Minister Nawaf Salam commented Tuesday on the resumption of Israeli-Lebanese negotiations in Washington by saying that "what remains is to consolidate a ceasefire throughout Lebanon."

The path of negotiation is "the least costly for the country, especially since wars throughout history have ended with a victor and a vanquished, or with negotiations to resolve the conflict," President Joseph Aoun said on Tuesday.

The Lebanese, Arab and international contacts that managed to avert an Israeli strike on Dahieh on Monday will contribute to a better atmosphere in today's talks between Israel and Lebanom in Washington, Baabda sources said on Tuesday.

The ambassadors of Israel and Lebanon began a new round of direct talks in Washington on Tuesday after U.S. President Donald Trump said he had received commitments from both sides to de-escalate.

Lebanon's health ministry said Tuesday that Israeli strikes a day earlier near a hospital in the southern city of Tyre killed four people and wounded 127 others, including 39 staff from the facility.

The ministry statement said the wounded staff at the Jabal Amel hospital were "four doctors, 27 nurses, and eight (administrative) employees -- four of whom are in critical condition and receiving treatment in intensive care".

Israel's defense minister said Tuesday Washington had given its backing to potential Israeli strikes on Beirut's southern Dahieh suburb if the Iran-backed group attacked northern Israeli communities.

"The Prime Minister and I led a coordinated move together with the IDF to establish a new equation... If Israeli towns continue to be attacked, we will evacuate and strike the Shiite Dahieh quarter in Beirut, Hezbollah's stronghold," Israel Katz said at a conference, a defense ministry statement said.

Ceasefires have been announced, often to great fanfare, in Gaza, Lebanon and Iran. So why is there still so much fighting?

In just the last few weeks, Israeli forces have captured more territory in Gaza and killed two top Hamas militants there, as well as more than a dozen other people. In Lebanon, Israeli troops captured a Crusader fortress over the weekend in their deepest incursion in 26 years, as Hezbollah kept up rocket fire into northern Israel.

Lebanon's public university said on Tuesday it was postponing some examinations after two students and their father were killed while returning to south Lebanon after sitting exams a day earlier.

The state-run National News Agency (NNA) said a dentist from the border village of Qlayaa "was killed with his son and daughter when an enemy drone targeted his car on the Nabatieh-Khardali road" on Monday.

Israel has portrayed the capture of a Crusader-era fortress as a turning point in its offensive against Hezbollah, but the operation has also sparked fears of a new quagmire in south Lebanon.

Israeli authorities praised the symbolism of the capture of Beaufort, perched atop a rocky outcrop, as they announced its capture in an assault documented by military drones.