The Israeli military said on Sunday that the brother of the man who drove a truck into a synagogue in Michigan last Thursday was a Hezbollah commander who was killed in an Israeli airstrike in Lebanon earlier this month.

The synagogue attacker, Ayman Mohamad Ghazali, was a naturalized U.S. citizen born in Lebanon. Four of his relatives were killed in an Israeli airstrike on March 5: his brother, Ibrahim, and his two children, as well as another brother, Qassem, according to a Lebanese official and a Muslim leader in Michigan.

Israel said it had confirmed the death of Ibrahim Mohamad Ghazali and his role in Hezbollah after an intelligence analysis but provided no details about how it had verified the information. It did not link the attacker in Michigan to Hezbollah.

A Hezbollah official, speaking on condition of anonymity to talk to the media, denied that Ibrahim or his family were affiliated with the group. He told The New York Times the Michigan attacker was motivated by revenge for his family.

The Israeli military described the target of the March 5 strike as a Hezbollah “military structure,” where it said weapons were stored and operatives of the militant group were present.

The military said Ibrahim had led a Hezbollah weapons team for the Badr unit, a division of the group’s southern command. It did not mention the deaths of any other family members.

The Lebanese official said the Israeli airstrike hit a three-story building and that Ibrahim’s wife was seriously wounded. The Israeli military and Lebanese news reports said the building was in the eastern town of Mashgharah.

The strike was part of an escalating cross-border battle between Israel and Hezbollah that began days after Israel and the United States attacked Iran on Feb. 28.

Hezbollah fired rockets into Israel on March 2 in support of its patron, Iran, opening up a new front in the expanding conflict. That prompted waves of devastating Israeli strikes on Lebanon and persistent rocket attacks from Hezbollah on Israel.

About 800 people have been killed in Lebanon, according to the country’s health ministry.

Missile fire from Iran has killed at least 12 people in Israel during the current conflict, according to the Israeli ambulance service.

The Israeli military said in its statement on Sunday that Hezbollah’s Badr Unit was “responsible for firing hundreds of rockets toward the citizens of Israel during the war.”

Mr. Ghazali, the synagogue attacker, had attended a memorial for his slain family members at mosque in Dearborn Heights, Michigan, on March 8, according to the imam there, Hassan Qazwini of the Islamic Institute of America.

On the morning of March 12, Mr. Ghazali rammed a truck through the door at Temple Israel in West Bloomfield Township and moved down a hallway. He exchanged gunfire with security guards after his vehicle became lodged in the hallway and he killed himself.

The vehicle was found to be loaded with fireworks. One guard was injured. Roughly 140 children and staff members at the temple’s preschool were safely evacuated.

U.S. officials called it a targeted act of violence against the Jewish community.

Hwaida Saad contributed reporting from Beirut.