Abdel Rahim al-Mismari was convicted for a 2017 shootout in the Western Desert between police and militants, which left 16 police dead.
Amir Mohammed Abdul Rahman al-Mawli al-Salbi had issued religious edicts to justify the persecution of the Yazidi minority, a campaign the UN has described as genocide.
The militia's ties to the Iranian regime have left the government with few options to access aid, which has fueled public anger against it.
The militia and its backer Iran have been depriving Lebanon of real solutions to resolve its suffocating financial crisis.
US sanctions targeting an IRGC-linked shipping network and other large global firms demonstrate a resolve to shut down Iran's oil trade.
The security chief and four of his aides were killed in their vehicle by an improvised explosive device while they were on duty in Hadramaut.
Illegal crossings with Syria that are controlled by Hizbullah have created both economic and security problems for Lebanon.
Mohammed al-Shamrani was radicalised at least as far back as 2015, and had been in contact with operatives from al-Qaeda's Yemen branch since then.
Germany's decision to ban all the militia's activities on its soil is the latest in a string of similar restrictions and may trigger further action.
Religious authorities and politicians denounced the attack and mourned the soldiers who were killed when their vehicle was targeted.