you thought wrong. Your supreme pontiff Gregory VII promoted the idea in 1074, before Alexis took the throne.
Precisely, and don't forget it was the Pope who called the Crusades. I thought that the First Crusade was called because the Orthodox emperor Alexios I asked Pope Urban II for mercenaries to help him resist the Muslims who were invading the Byzantine Empire.
Gregory did nothing more than to show his love and compassion for the Orthodox Christians who were being overrun:
"Gregory, bishop, servant of the servants of God, to all who are willing to defend the Christian faith, greeting and apostolic benediction.
We hereby inform you that the bearer of this letter, on his recent return from across the sea [from Palestine], came to Rome to visit us. He repeated what we had heard from many others, that a pagan race had overcome the Christians (Greeks) and with horrible cruelty had devastated everything almost to the walls of Constantinople, and were now governing the conquered lands with tyrannical violence, and that they had slain many thousands of Christians as if they were but sheep. If we love God and wish to be recognized as Christians, we should be filled with grief at the misfortune of this great empire [the Greek] and the murder of so many Christians (Greeks). But simply to grieve is not our whole duty. The example of our Redeemer and the bond of fraternal love demand that we should lay down our lives to liberate them. "Because he has laid down his life for us: and we ought to lay down our lives for the brethren," [1 John 3:16]. Know, therefore, that we are trusting in the mercy of God and in the power of his might and that we are striving in all possible ways and making preparations to render aid to the Christian empire [the Greek] as quickly as possible. Therefore we beseech you by the faith in which you are united through Christ in the adoption of the sons of God, and by the authority of St. Peter, prince of apostles, we admonish you that you be moved to proper compassion by the wounds and blood of your brethren and the danger of the aforesaid empire and that, for the sake of Christ, you undertake the difficult task of bearing aid to your brethren [the Greeks]. "
Nothing happened until the Orthodox emperor Alexios begged the Pope of Rome, Urban II, to send in an army to help him. The Pope, having mercy and sympathy and great affection and love for the Orthodox people of the Byzantine empire, went way out of his way to send Roman Catholic soldiers to die for the Orthodox of the Byzantine empire at the request of the Byzantine Orthodox emperor Alexios.