I followed your tip. Maybe I did forget to tell you I am using the travelia template. So duplicating and assigning the Helix ultimate template completely broke my site.
I have duplicated now the travelia template, but it still does not work.
Problem seems to be that the template (which is default for all pages) cannot be changed. So I recreated 3 templates, one for each language in addition to the default for all languages .
After that I clicked on the default star in the list of templates, making the new French template default. And now I cannot change that one anymore to make it default for only French pages...