Hún gleymdi að læsa bílnum, en tók eftir því áður en hún fór inn.

In Icelandic, gleyma often takes an infinitive with að when you mean to forget to do something.

• gleyma að læsa = to forget to lock
• gleyma að hringja = to forget to call

So:

• Hún gleymdi að læsa bílnum = She forgot to lock the car

The verb gleymdi is the past tense of gleyma.

Because it comes after að and expresses the action that was forgotten.

• að læsa = to lock

This is very similar to English forgot to lock. The first verb is finite:

• gleymdi = forgot

and the second verb stays in the infinitive:

• að læsa = to lock

Because the verb læsa takes the dative case in Icelandic.

So:

• bíll = car (dictionary form, nominative singular)
• bílinn = the car (accusative singular)
• bílnum = the car (dative singular)

Since læsa governs the dative, Icelandic uses:

• læsa bílnum = lock the car

This is something learners often just have to memorize with the verb:

• læsa + dative

Literally, it means something like took after that, but as an expression it means:

• to notice it
• to realize it

The verb phrase is:

• taka eftir einhverju = to notice something

It also requires the dative case after eftir.

So:

• því is the dative form of það = it / that

Therefore:

• tók eftir því = noticed it / realized it

Because the expression taka eftir requires the following word to be in the dative.

The pronoun það changes by case:

• nominative: það
• accusative: það
• dative: því
• genitive: þess

Since eftir here takes the dative in this expression, Icelandic uses:

• eftir því = it / that in the dative

So tók eftir því is grammatically required, not optional.

Because Icelandic often leaves out the subject in the second part of a sentence when it is the same as in the first part.

So this sentence means:

• Hún gleymdi að læsa bílnum, en (hún) tók eftir því...

The second hún is understood, so it does not have to be repeated.

This is similar to English:

• She forgot to lock the car, but noticed it before she went inside.

English can also drop the second she in this kind of structure.

En means but.

It connects two contrasting ideas:

• Hún gleymdi að læsa bílnum = She forgot to lock the car
• en tók eftir því áður en hún fór inn = but noticed it before she went inside

So en shows contrast: she forgot, but then she realized it in time.

Áður en means before when it introduces a clause.

Examples:

• áður en hún fór inn = before she went inside
• áður en ég borða = before I eat

It is a very common structure in Icelandic.

So in this sentence:

• áður en hún fór inn = before she went inside

Because the whole sentence is describing past events.

• gleymdi = forgot
• tók eftir = noticed
• fór inn = went inside

The sequence is:

1. she forgot to lock the car,
2. she noticed that,
3. this happened before she went inside.

Since all of it happened in the past, fór is past tense.

Fara means to go, and inn means in / inside.

Together:

• fór inn = went in / went inside

Icelandic often uses these small direction words very naturally:

• inn = in, inside
• út = out
• upp = up
• niður = down

So hún fór inn is a normal way to say she went inside.

This is a subordinate clause introduced by áður en.

The normal order here is:

• áður en
  • subject + verb + adverb/direction

So:

• áður en hún fór inn

Word by word:

• áður en = before
• hún = she
• fór = went
• inn = inside

This is straightforward subordinate clause word order.

It is more like noticed it, realized it, or became aware of it.

It does not mean remembered in the usual sense.

So here the idea is:

• she forgot to lock the car,
• then she realized that fact before going inside.

A good English translation would be:

• but noticed it before she went inside
• but realized it before she went inside

Yes, very often.

This sentence is a good example:

• gleyma can involve a certain structure: gleyma að + infinitive
• læsa takes the dative
• taka eftir takes the dative

So it is very useful to learn vocabulary like this:

• læsa + dative
• taka eftir + dative
• gleyma að + infinitive

That will help you build correct sentences much more easily.