Israel's Chief of Staff Gadi Eisenkot rejected Hezbollah's account of Badreddine's death at a security conference on Tuesday.
"Those findings show to what degree relations between Hezbollah and its patron Iran are tense and complex". "No more, I say from now that Israel's policy is 'Lebanon is Hezbollah and Hezbollah is Lebanon", the education minister says.
He also apparently directed Hezbollah's ground offensives in al-Qusair, a Syrian border town, in 2013 as part of the Shi'ite group's support for the Assad regime in Syria.
An Israeli military official told the AP news agency that Israel believed the order to kill Badreddine, known by the nom de guerre "Zulfikar", was given by Hassan Nasrallah.
Israeli intelligence believes Badreddine had been feuding with Iranian military commanders in Syria. He said the death of Badreddine illustrated "the depth of the internal crisis within Hezbollah".
He was one of five Hezbollah members indicted by the UN-backed Special Tribunal for Lebanon in the 2005 in connection to the killing of former Lebanese Prime Minister Rafik al-Hariri and 21 others.
Lieutenant-General Gadi Eisenkot, chief of the Israeli armed forces, said that Israeli intelligence had corroborated reports of Hezbollah assassinating one of its own commanders, but did not elaborate on the circumstances.
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Badreddine was killed in May 2016 in a mysterious explosion near Damascus shortly after a meeting with his commanders.
The Lebanese-based Hezbollah is a sworn enemy of Israel, and has fought the state in a month-long war in 2006.
Hezbollah said Badreddine was a "great jihadi leader", and blamed Syrian rebels for his death, although Hezbollah MP, Nawar al-Saheli, attributed the attack to Israel, and vowed that "the resistance will carry out its duties at the appropriate time".
Israel is not actively involved in the Syrian fighting.
"In the past three weeks, we have heard about how their Hezbollah commander was killed. according to a wave of publications, he was killed by his own commanders". It gave no cause for the crash and said the incident was being investigated, but said there was 'no risk of a breach of information'.
Hezbollah's media centre published photographs of what it said was a drone it had shot down in Syrian airspace in the Golan Heights.