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This is just the smallest thumbnail sketch of an outline that may give some bare bones for you to hang many more details on regarding how the Eastern Church became 'Eastern Orthodox'. Although the actual, official split between East and West did not happen until 1054, disagreements between the Eastern and the Western areas of Catholicism began hundreds of years earlier. Here is the background to that.
Constantine the Great established a new capital on the Bosphorus, as a counterpart to Rome, and called 'The New Rome' - Constantinople. This became the home of the Eastern Christian tradition. The Eastern Church had its 'Byzantine Fathers'. Some of its bishops became recognized as 'chief metropolitans', also known as 'patriarchs'. The 'Ecumenical Patriarch' became the spokesman for Eastern Christianity. Byzantium expanded but during the 700s suffered much loss of territory, many of its people emigrating from the East in face of threats from Persians and Arabs.
Emperor Leo III (717-41) launched an attack on the use of icons in 730. This caused eruptions. Here is a quote from the history book I am using to cull this information from:
"The patriarchal Bishop of Rome dared to condemn the destruction of icons. The Emperor retaliated by removing Sicily, southern Italy and the entire western part of the Balkans and Greece from the patriarchate of Rome and into the patriarchate of Constantinople. This, more than anything else, forced the bishop of Rome to seek the support and protection of the Franks. Under Leo III, and his son Constantine V (741-75), those supporting icons were persecuted vigorously. A synod of 338 bishops met at Constantinople in 754 and described all use of icons as idolatry. All remaining icons must be destroyed. Supporters of icons were excommunicated, mutilated and sent into exile." The History of Christianity, pp. 245-8, Lion 1985 reprint
When the patriarchal bishop of Rome forged an alliance with the Frankish kings, and in 800 crowned Charlemagne 'Emperor of the Romans', then for all practical purposes, a separate Roman Empire in the West was created.
This iconoclastic dispute was finally settled by a synod in 843, when Patriarch of Constantinople John Grammaticus was deposed. The ruling of the seventh council was confirmed and Orthodox churches today still celebrate the first Sunday in Lent each year as the 'feast of Orthodoxy' to commemorate the end of the iconoclastic controversy.
The Eastern Churches identify themselves with the first seven ecumenical councils (from 325 to 787), referring to themselves as "The Church of the Seven Councils".
It was later disagreement about wording for the role of the Holy Spirit in the Trinity that led to the 1054 split from Rome. Eastern Christianity survives today in the Orthodox churches.