I hope I got the terminology right, here is the case I want to ask about:
In an exercise, I have to find out what the protagonists say in which situation. The situation is described in a short sentence. Here is the sentence in question:
社員研修の制度が利用できないか聞くとき "When he asks if it wouldn't be possible to use the employee training program..."
What bothers me is the sole か. If I remember correctly, for indirect/embedded questions, か alone is used only if a question word is contained in the respective clause, like なん or どう. I can't see something like that being the case here, so I'd expect か どうか here. My textbook never taught me wether this "rule" gives you any leeway in omitting the どうか even though there's no such question word. Or maybe I misinterprete something here.