Directly changing the pronoun "dir" to the formal "Ihnen" in "lass dir Zeit" would give:
lasst Ihnen Zeit
but this is incorrect, what is correct is
lassen Sie sich Zeit
whereas "lass du dich Zeit" is incorrect (is it?).
Is the above correct? Why does the reflexive "sich" appear in the formal case but not in the informal one?
This follows from how reflexive pronouns are used when a verb is conjugated in German. Here's one verb with accusative and one with dative reflexive pronoun:
Ich freue mich. Ich mache es mir bequem.
Du freust dich. Du machst es dir bequem.
Er/sie/es freut sich. Er/sie/es macht es sich bequem.
Wir freuen uns. Wir machen es uns bequem.
Ihr freut euch. Ihr macht es euch bequem.
Sie freuen sich. Sie machen es sich bequem.
So in third person singular and plural, the reflexive pronoun to use is always "sich", while in first and second person, other than in English, the "normal" accusative or dative pronouns are used (i.e. the ones that would also be used if the subject was a different person). This is possible because they are unambiguous in first and second person but not in third person. E.g., in third person plural, "Sie machen es ihnen bequem" (They make them comfortable) and "Sie machen es sich bequem" (They make themselves comfortable) are two different things.
The formal "Sie" address is grammatically the same as the third person plural, and that's why it uses sich:
Sie lassen sich Zeit. (They are taking their time or You are taking your time)
Or in imperative:
Lassen Sie sich Zeit!
Whereas in second person singular, as seen in the conjugation above, the "normal" pronoun dir is used:
Lass (du) dir Zeit!
The verb is "jemandem Zeit lassen", and of course it can be used reflexively as "sich Zeit lassen" if the subject of the sentence is allowing time for themselves.
Deriving from that the other form follow, especially also imperative ones. In the imperative the subject you address usually is omitted in the 2nd person (you as singular or plural: Du or Ihr). It can be added, but is uncommon. Yet with the 3rd person imperative you cannot leave out the subject of the sentence. Either case, the reflexive pronoun must match the subject:
Notably the imperative in the first person, especially in singular, is of course at least very uncommon and usually formulated as statement "Ich lass mir Zeit", while in the 1st person plural it is indeed found as request to the group you are part of.
But of course it need not be used reflexively, but still the Dativ object requires the appropriate pronoun, if a pronoun is used:
etc etc.
Pretty good question. Yes the correct form would be "Lassen Sie sich Zeit". Both "lasst ihnen Zeit" and "Lass du dich Zeit" would sound very wrong for the purpose.
"Lasst ihnen Zeit" would also do nothing in terms of politness but rather be confusing in terms of singular and plural. as "Lasst" would imply that you command or beg a group of people (imperative plural) and "ihnen" would imply that you're talking about a group of people. So it would technically be correct from a syntactic point of view but the meaning would be a very different.
And "Lass du dich Zeit" sounds very wrong. If you had to, you could say "Lass du dir Zeit (damit)" oder "Du kannst dir (damit) Zeit lassen" but that also doesn't sound right (EDIT:That would actually overemphasize the "du", like in "YOU (in particular) can take your time"; so maybe it's fitting as the "Sie" also emphasizes the person you're talking to. Though it's weird to be both informal and so specific about the person) and at best old fashioned, the more natural Form would be to shorten it to "Lass dir Zeit (damit)". It's also more general and not specific to a particular task (unless obvious from context).
I Think it is an Akkusativ question:
"lassen Sie" > was? = Zeit > wen? = sich
If we say "lassen Sie Ihnen Zeit" (what could be correct in other context), we would refer to a 3rd group of people which would become this "Zeit", as "Ihnen" in this phrase would refer to somebody else as "Sie".
So, if we want that "Sie" gives the "Zeit" to itself, we need this reflexive "sich".
