A false awakening IS just a regular dream. It's just a regular dream where the situation is that you are dreaming that you just woke up. Most likely you'll be in your bedroom or another place where it is common for you to sleep/wake up. There's no difference, at least that I have ever heard of, between this and other dreams. A problem with FAs is that you are in your room, so probably not by default somewhere more interesting. Other than that, yes, they can be full on amazing lucid dreams just like any other lucid dream.