It is finally working ! 
Thank you very, very much Zoltan ! 
I will soon modify this message and give both the AX12 library and SerialHalfDuplex . Do you know where I can place these files so that the community can use them?
Hello Mickaël,
I’m glad you made it work! Although I have never done it, perhaps the best is to make a pull request on the original library and then publish your version for Mbed Os 6.