Quick corrections, if you don't mind.
"Lucifer" doesn't exist in Islam; "Lucifer" isn't Satan's name, but "Iblis".
And Satan isn't an angel in Islam; he's a jinn.
And Satan didn't "rebel" in Islam: he disobeyed. He refused to prostrate to humans, even under God's direct command, seeing himself as better than humanity.*
*Source:
Walaqad khalaqnakum thumma sawwarnakumthumma qulna lilmala-ikati osjudoo li-adamafasajadoo illa ibleesa lam yakun mina assajideen
And We have certainly created you, [O Mankind], and given you [human] form. Then We said to the angels, "Prostrate to Adam"; so they prostrated, except for Iblees. He was not of those who prostrated.
Qala ma manaAAaka allatasjuda ith amartuka qala ana khayrun minhukhalaqtanee min narin wakhalaqtahu min teen
[ Allah ] said, "What prevented you from prostrating when I commanded you?" [Satan] said, "I am better than him. You created me from fire and created him from clay."
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Qur'an: al-A'raf (Surah 7), verses 11-12, Sahih International Translation [Linked]
Note: Iblis is not taken as being an angel, even though he is "with the angels". Angels have no free will in Islam. Jinn, of which Iblis is supposed to be, have free will like humans. The fact that Iblis refuses, shows that, according to Islamic beliefs, shows he has free will, and therefore he cannot be an angel.
Not Muslim, by the way. If I've made any mistakes, hopefully a Muslim will spot them and correct me.