اليوم أكتب الواجب في المكتبة قبل أن أعود إلى البيت.

A natural breakdown is:

• اليوم = today
• أكتب = I write / I am writing
• الواجب = the homework / the assignment
• في المكتبة = in the library
• قبل أن = before
• أعود = I return
• إلى البيت = to the house / home

So the literal order is roughly:

Today I write the homework in the library before I return to the house/home.

Because Arabic word order is flexible, and putting اليوم first gives it a little emphasis, like:

• As for today, I’m writing the homework...
• Today, I write the homework...

It is not the only possible order. You could also say:

أكتب الواجب اليوم في المكتبة قبل أن أعود إلى البيت.

Both are understandable, but starting with اليوم highlights the time.

It is built into the verb أكتب.

In MSA, verbs usually show the subject, so أكتب already means I write or I am writing. You do not need to add أنا unless you want extra emphasis.

• أكتب = I write
• أنا أكتب = I write / I am writing, with emphasis on I

So Arabic often leaves out the separate subject pronoun when the verb already makes it clear.

أكتب is in the imperfect form, which often covers meanings that English splits into several tenses.

Depending on context, it can mean:

• I write
• I am writing
• I will write

In this sentence, اليوم helps set the time frame, so the intended meaning is something like today I am writing or today I will write. Arabic often relies on context more than English does.

Because الواجب means the homework / the assignment, while واجب would mean a homework task / a duty / an assignment in a more indefinite sense.

In many contexts, homework is treated as something known or specific, so the definite article الـ is natural.

• الواجب = the homework
• واجب = a duty / an assignment

Also, in classroom language, الواجب is a very common word for homework.

Because في means in.

So:

• في المكتبة = in the library

This is a prepositional phrase. After a preposition like في, the noun is grammatically in the genitive case. In fully vowelled Arabic, it would be:

في المكتبةِ

In normal writing, the case ending is usually not written, so learners just see في المكتبة.

قبل means before, and أن introduces a following verb clause.

So:

• قبل أن أعود = before I return

In standard Arabic, this is a very common structure:

• قبل أن + verb

Without أن, you usually would not say this same idea in the same way. If you wanted to avoid أن, you would normally change the structure, for example:

• قبل عودتي = before my return

So قبل أن أعود is the normal MSA way to say before I return.

Yes. After أن, the verb is normally in the subjunctive mood in MSA.

So grammatically:

• أعودُ = indicative
• أعودَ = subjunctive, after أن

In ordinary Arabic writing, short final vowels are usually not written, so both look the same as أعود. But if the sentence were fully vowelled, it would be:

قبلَ أن أعودَ

So the grammar changes, even though the spelling usually does not.

Because إلى means to / toward, and it shows movement to a destination.

• أعود إلى البيت = I return to the house / go back home

If you removed إلى, the sentence would not sound correct in standard Arabic for this meaning.

Also, Arabic often uses the house where English simply says home. So إلى البيت is very natural even though English usually would not say to the home here.

Arabic often uses the definite article in places where English uses no article.

So:

• إلى البيت very often means simply home

This is a good example of Arabic and English not matching word for word. Even though البيت literally means the house, in context it often works like English home.

If fully vowelled in careful MSA, the sentence would be:

اليومَ أكتبُ الواجبَ في المكتبةِ قبلَ أن أعودَ إلى البيتِ

Here is why:

• اليومَ: accusative as an adverb of time
• أكتبُ: indicative verb
• الواجبَ: accusative because it is the direct object
• المكتبةِ: genitive after في
• قبلَ: usually accusative in this use
• أعودَ: subjunctive after أن
• البيتِ: genitive after إلى

In normal everyday writing, these final short vowels are usually omitted.

Yes. Arabic allows several natural orders, depending on emphasis.

For example:

• اليوم أكتب الواجب في المكتبة قبل أن أعود إلى البيت
• أكتب الواجب اليوم في المكتبة قبل أن أعود إلى البيت
• في المكتبة أكتب الواجب اليوم قبل أن أعود إلى البيت

All of these are possible, but they emphasize slightly different parts. The original sentence is natural and gives early emphasis to today.

Yes.

The past-tense form is:

• عاد = he returned

From that verb, the imperfect forms include:

• أعود = I return
• تعود = you return / she returns
• يعود = he returns
• نعود = we return

So in this sentence, أعود is the I form of the verb عاد.