A lucid dream within a non-lucid dream

I have some Inception **** going on here, I was wondering if it happens to anyone else. It's happened to me several times now.

Sometimes I begin daydreaming within a non-lucid dream, and that daydream behaves pretty much exactly like a lucid dream, but my dream self thinks I'm just daydreaming in real life. So it's not exactly a lucid dream, because I treat it like a daydream and don't understand that it's actually a dream within a dream. I don't do reality checks, don't try to stabilize the dream, or anything like that. But at the same time, I do all the kinds of things I do in LDs (or daydreams), such as flying, being a knight, whatever, and it's a lot more vivid than just a daydream. I have full control over the daydream and it's very much like an LD, but in the dream I believe it's a different type of imagination-scape (a daydream). Usually, I'll stop daydreaming at one point and continue the upper-level dream as a non-lucid dream. Other times something happens that lets me in on the fact that I'm actually dreaming. For example sometimes I'll take something out of the daydream and into the upper-level dream and realize that that's a little fishy.