Complex sentence: After Oxford, he began teaching Classics at Eton and he remained there until 1963, when he became headmaster at Clifton College; he remained at Clifton until 1975 when he became the head of the Pilgrims School, Winchester.
Split into simpler sentences: After Oxford, he began teaching Classics at Eton and he remained there until 1963 when he became headmaster at Clifton College. He remained at Clifton until 1975 when he became the head of the Pilgrims School, Winchester.