While creating an entry for a chapter, one can also create an editor for a book, which contains the chapter (see the screenshot attached). An entry of the editor requires the last and the first name. The entry form for the first name is disabled, so that you can't type anything in it.
We have found out, that in the HTML for the "first name" entry form there is an attribute disabled="disabled". That seems to be the reason for the inaktive entry field. We have removed the attribute from the HTML and it works now properly.