I just found out that LT uses internally de_DE_frami and does what I have suggested: LT renames de_DE_frami files in de_DE files. Obviously, when using LT, the spelling check is performed twice. Once from TXS and once from LT. This does not seem to me a good idea. In addition, supplements to the de_DE.ign file are only accepted by TXS, but not by LT. These changes must be copied manually into the files ingnore.txt or spelling.txt. Since LT runs on my OS X notebook, I have significantly more program...
Problem solved. On OS X I work with the de_DE_frami dictionary. When I switch to TeXstudio de_DE dictionary, then LT works. I use the de_DE_frami dictionary because it has almost twice as many words as the de_DE dictionary of TeXstudio. The solution is that the de_DE dictionary files are deleted and the de_DE_frami.x files are renamed to de_DE.x.
In addition, I would like to add that the spelling check also does not work. What I see in the OS X installation is the internal spelling check of TexStudio. In the Windows installation, I changed the display of the spelling errors and found that LT also displays spelling errors.
TexStudio 2.12.6 does not mark grammar errors