Since the Church allows the Traditional Rite to be celebrated in accordance with Ecclesia Dei, the adherence to the "Traditional Rite" is obviously NOT the reason why the SSPX is in schism.
“There is also the highly debated act of the consecration of 4 bishops by the Archbishop Marcel Lefebrve in 1988 which some contest lead to automatic excommunication the highest punishment the Church can bestow.”
Quoting myself — from my own article.