Have you ever realized in a dream that you are sleeping? And, what do you think, does this speak about any peculiarities of the character / mind / etc.?

It happens, and very often. For this I did nothing. OS begins when I can take control in a dream such a thing as a feeling of pressure from the outside (in my case - a strange feeling when someone pushes you hard diagonally, I don't know how to explain it better), or I hear own cry. But I do not think that this speaks about any unique and special traits / skills of a person.

Just one of the abilities of the human brain, nothing more.

Happened and repeatedly. The first lucid dreaming began after meeting Castaneda. It's just that one day I was able to look at my hands in a dream.
In this matter, practice is really necessary, awareness does not come once and for all.
The only thing this skill turned out to be useful was the ability to prevent a nightmare by changing the course of events. Actually, now awareness comes only in the tense moments of sleep and then quickly escapes me again.

It happened, but right after a few seconds of OS I woke up, without training I can't keep it for a long time: c

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This is called Lucid Dreaming (OS). It is almost impossible to hold them for more than a few seconds (local time) without training and preparation. During OS, you can also change sleep.

They happen during normal sleep, very rarely (in the majority). They occur during the phase of light sleep, when, let's say, the connection between reality and sleep is the strongest.

The fact that you had a lucid dream for several seconds does not describe or characterize you in any way, does not give any cool skill ... But how will the achievement go.