I had a professor who told me during the year that he'd be happy to write me a letter of recommendation for a research degree I was contemplating applying to. I've since decided to apply.
My university (in Europe) is on break until 1 October. I emailed the professor asking for the letter three weeks ago on his university email address. He hasn't answered me. However, I sent him a connection request on LinkedIn, and he accepted it within 45 seconds. I am thus wondering if he is not checking his university email during our vacation. However, I would like to know if he is willing to write it before October, as waiting until then would give me insufficient time to request one from someone else.
Would it be inappropriate for me to send him a follow-up letter request via LinkedIn, since I know he's checking that? Would it be more appropriate to make reference to the email I sent him, or send my entire request via the messenger? Or should I limit myself to communication via the university email address and just hope he reads it?