grammar - What's the difference between ～てください and ～ていてくださ い?

This question has come up on the side of at least a couple of other questions, but I don't think it's been definitively answered, so let's see if we can come up with something solid.

After going through the first ten pages of Google results for both "ていてください" and "ていて下さい" (very scientific, I know), I found 43 verbs and expressions that precede these forms. The most common:

待つ (まつ):                50 results

見る (みる):                33
楽しみにする (たのしみにする):  25
覚える or 憶える (おぼえる):  19
生きる (いきる):              9
期待する (きたいする):         5
見守る (みまもる):            4
強いる (しいる):              3
離れる (はなれる):            3
知る (しる):                 3
思う (おもう):                3

待つ is somewhat expected, but there are some interesting entries here, such as 生きる and 離れる. How is 生きてください different from 生きていてください, and likewise, how is 離れてください different from 離れていてください? At a minimum, let's try to answer:

1. What is the difference in nuance between these two?


2. When should you choose one over the other?

Bonus points if you can draw from a trusted reference.

Related:

Answer

I think each verb is somewhat a case of its own, but generally speaking they all seem to relate somehow to the progressive nature of the ～ている form.

If we get to the specifics, here are my impressions, based on my experience, intuition and grammatical understanding (all of them seem to point to the same thing in this case, which is good). Sorry, no references now, but maybe I'll try to search for some academic papers later. There's a good chance you can find some on Google Scholar:

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  生きていてください
  I'm almost positive I've already heard or seen it somewhere. It seems to mean something along the lines of "Please keep being alive" or "Please stay alive", instead of 生きてください which means just "Please live", with an implied meaning such as "living through something" or "choosing life over death", but not "staying alive". According to Google, it's actually much more common than 生きてください, and that's only natural, since in English too you'd usually ask people to stay alive instead of just "live". :)

  
  


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  離れていてください
  This is again the case of a request for keeping some state for a prolonged period instead of just doing a single action. I don't think I've encountered this specific verb in this form (though it does seem very common according to Google), but it's easy to understand it as "separate and remain separated", or maybe even more simply "stay away", "keep distance".

  
  
  


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  覚えていてください
  Remember for a prolonged period. Or as we'd probably say in English: keep in mind.

  
  


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  見守っていてください
  Observe closely for a prolonged period. Or idiomatically: Keep an eye.

  
  


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  期待していてください Expect for a prolonged period. Or idiomatically: stay tuned.

  
  

You've probably noticed how much I've had to resort to idioms in English to translate these Japanese phrases more accurately. That's because English doesn't have an progressive imperative construction. And you may have also noticed I heavily used the verbs keep and stay in most of the idioms I gave. That's because while English cannot express a progressive imperative, it can always express a meaning of remaining in a certain state by using verbs such as stay, remain and keep. These verbs usually take an adjective or an adverb (including a preposition-as-adverb) and not another verb, so their use is very idiomatic and not as regular as in Japanese.