nprofile1qyt8wumn8ghj7un9d3shjtnyd968gmewwp6kytcqypren5k53ms4x26agn2qg4anqpe3a7zuy39qjjd5slavzskqn2qgxcaztfm (nprofile…ztfm) "It seems to me like one can easily have such a setup where x doesn't come from the subfield k, namely take i=j."
Slightly less trivially, consider i, j : k(x, y) -> k(x, z, t) that send x to x and y to z or t respectively. I don't know about the rest.