Sie steckt den Schlüssel in den Ärmel ihres Mantels, damit sie ihn nicht verliert.

Here steckt comes from stecken. In this sentence, it means something like puts, sticks, or tucks into a place.

So Sie steckt den Schlüssel in den Ärmel ihres Mantels means she is tucking/putting the key into the sleeve of her coat.

This is a very common verb when something is being put into a narrow or enclosed place:

• etwas in die Tasche stecken = to put something into a pocket
• etwas in die Schublade stecken = to put something into the drawer
• den Schlüssel in den Ärmel stecken = to tuck the key into the sleeve

Because den Schlüssel is the accusative case, which is used here for the direct object of the verb.

The verb stecken is acting on the key. She is putting the key somewhere, so the key is the direct object.

• nominative: der Schlüssel = the key
• accusative: den Schlüssel = the key

Since Schlüssel is masculine, der changes to den in the accusative.

Because this sentence expresses movement into the sleeve, not location inside it.

German prepositions like in can take:

• accusative for direction/movement toward a destination
• dative for location/static position

Here, the key is being moved into the sleeve, so German uses the accusative:

• in den Ärmel = into the sleeve

If the key were already there and you were only saying where it is, you would use the dative:

• Der Schlüssel ist im Ärmel. = The key is in the sleeve.

So:

• in den Ärmel = into the sleeve
• im Ärmel = in the sleeve

ihres Mantels is a genitive phrase meaning of her coat.

So:

• der Ärmel ihres Mantels = the sleeve of her coat

This shows possession:

• der Ärmel = the sleeve
• ihres Mantels = of her coat

A learner might expect something like von ihrem Mantel, and that is possible in some contexts:

• der Ärmel von ihrem Mantel

But ihres Mantels is more standard and more elegant in written German.

Also note the endings:

• ihr becomes ihres in the genitive singular here
• Mantel becomes Mantels in the genitive singular

Here damit means so that or in order that.

It introduces a clause expressing purpose:

• Sie steckt den Schlüssel in den Ärmel ihres Mantels, damit sie ihn nicht verliert.
• She puts the key in the sleeve of her coat so that she does not lose it.

So the first part is the action, and the damit clause gives the reason or goal behind that action.

Because damit introduces a subordinate clause, and in German subordinate clauses the conjugated verb usually goes to the end.

Main clause:

• Sie steckt den Schlüssel ...
• The verb steckt is in second position.

Subordinate clause:

• damit sie ihn nicht verliert
• The verb verliert goes to the end.

This is a very important German word-order pattern:

• weil sie müde ist
• dass er heute kommt
• damit sie ihn nicht verliert

ihn refers to den Schlüssel.

Since Schlüssel is masculine, the accusative pronoun is ihn:

• der Schlüssel → ihn

So:

• damit sie ihn nicht verliert = so that she does not lose it
• Here, it = the key

In normal German word order, the pronoun object ihn usually comes before nicht in this kind of clause.

So the natural order is:

• damit sie ihn nicht verliert

This means:

• sie = subject
• ihn = object pronoun
• nicht = negation
• verliert = verb at the end

Placing nicht before ihn would sound unusual here and would change the emphasis.

Here nicht negates the verb phrase verliert:

• sie ihn nicht verliert = she does not lose it

So the idea is not not the key, but not lose the key.

German often places nicht before the element it most directly negates, and in a subordinate clause with the verb at the end, nicht often appears right before that final verb.

Not by itself. Sie can mean:

• she
• they
• you (formal)

But in this sentence, the most natural meaning is she, especially if that is the meaning already given in your learning material.

Why?

• ihres Mantels suggests one person’s coat
• damit sie ihn nicht verliert most naturally means so that she doesn’t lose it

So although Sie can be ambiguous in isolation, the context strongly points to she here.

It can look confusing at first, but it is normal German.

The first Sie is the subject of the main clause:

• Sie steckt den Schlüssel ...

The second sie is the subject of the subordinate clause:

• damit sie ihn nicht verliert

In English we also often repeat the subject:

• She puts the key in her coat sleeve so that she doesn’t lose it.

So German is doing the same thing here.

German often uses the present tense where English may use either the present or will/won’t.

So:

• damit sie ihn nicht verliert

literally looks like:

• so that she does not lose it

But in natural English, that may be translated as:

• so that she won’t lose it

German does not need a future form here. The present tense already works for the intended meaning.

Yes, and that is a very useful question.

German often uses um ... zu when the subject of both clauses is the same.

Here the subject is the same person in both clauses, so you could also say:

• Sie steckt den Schlüssel in den Ärmel ihres Mantels, um ihn nicht zu verlieren.

This means the same thing: She puts the key in the sleeve of her coat so as not to lose it.

Difference:

• damit
  • full clause = very common, explicit
• um ... zu
  • infinitive = often more compact

Both are correct here.

It is singular here:

• der Ärmel = the sleeve
• den Ärmel = the sleeve (accusative)

The plural is also die Ärmel, so learners sometimes get confused.

In this sentence, you can tell it is singular because:

• the article is den, not die
• the phrase means into the sleeve, not into the sleeves

There is one — it is built into ihres Mantels.

English says:

• the sleeve of her coat or
• her coat sleeve

German says:

• den Ärmel ihres Mantels

So ihres carries the meaning of her. German is just expressing the possession in a different structure:

• the sleeve of her coat

Yes, it is natural and grammatically correct.

It sounds like normal German for describing someone putting an object somewhere safe so they do not lose it.

A slightly simpler everyday variant might be:

• Sie steckt den Schlüssel in ihre Manteltasche, damit sie ihn nicht verliert. = She puts the key in her coat pocket so that she doesn’t lose it.

But the original sentence is perfectly fine.