Lt. Gen. Hussein Ishaq died Saturday as rebels attacked a Syrian air defense base near the town of Mleiha, the official said. He spoke on condition of anonymity because he wasn't authorized to brief journalists about Ishaq's death.
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The Britain-based Syrian Observatory for Human Rights also confirmed Ishaq's death. It said he died from wounds he suffered during Saturday's fighting.
The Observatory, which bases its reports on a network of activists on the ground, said Ishaq was killed in clashes with fighters from the Nusra Front, the al-Qaida affiliate in Syria, and other Islamic rebel groups.
"We announce good news to the Islamic nation, of the killing of one of the leaders of unbelief, General Hussein Yaqoub Ishaq, head of the Air Defence Administration in Mleiha," the Islamic Front said in a statement.
Heavy fighting has happened around Mleiha in recent weeks. Forces loyal to Syrian President Bashar Assad have conducted a heavy bombing campaign around the area, which rebels see as strategically important due to its proximity to the capital.
Ishaq is one of a few high-ranking military officers to be killed in the Syrian war. It's unclear if his death will have any wider strategic or military impact on the war itself, however.
Reuters contributed to this report.