"I congratulate all the mujahedeen on different fronts and all Muslims for the victories won by our brothers in Iraq against the puppets (of Shiite Iran)," ideological leader Ibrahim al-Rubaish says in a video posted online.
Rubaish is considered to be the religious affairs chief of Al-Qaeda in the Arabian Peninsula (AQAP), the Yemen-based franchise of the jihadist network, formed in a 2009 merger of its Saudi and Yemeni branches.
"Who does not welcome the victories of Sunnis and the defeat of (Nuri al-) Maliki's gangs that have mistreated the Sunnis?" he asks of Iraq's controversial Shiite premier who continues to defy the president and international pressure to quit.
Maliki's policies in Iraq were widely blamed for widening sectarian divisions, causing a lack of confidence in the government and contributing to the rise of the Islamic State jihadists.
However, Rubaish held back from giving his support to Baghdadi, who at the end of June proclaimed the establishment of a caliphate straddling Iraq and Syria, with himself as "leader for Muslims everywhere".
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The AQAP chief instead urged "an end to infighting between (Sunni) mujahedeen", and called for "a common front to battle the enemies of our community, the puppets of (Shiite) Iran".
The Al-Qaeda branch in Yemen is considered by the United States to be the deadliest franchise of the extremist network.
AQAP remains faithful to Ayman al-Zawahiri, the Egypt-born successor to Al-Qaeda founder Osama bin Laden.
Al-Qaeda in the Islamic Maghreb (AQIM), another franchise in the network, last month rejected the declaration of an Islamist caliphate in Iraq and Syria, saying it had "defects" that jihadist leaders should rectify.
AQIM also reiterated its allegiance to Zawahiri.