Um leið og við komum heim skrifa ég stutta athugasemd um lagið og regnbogann í dagbókina mína.

It is a very common expression meaning as soon as, just as, or at the same time that.

Word-for-word, um leið is roughly at the same moment / simultaneously, and og introduces the clause that follows. In practice, though, it is best to learn um leið og as a fixed chunk.

So here it introduces a time clause: as soon as we come/get home.

Because Icelandic follows the V2 rule in main clauses. That means the finite verb normally comes in the second position.

Here, the sentence begins with the time clause Um leið og við komum heim. Since that whole clause takes the first slot, the verb of the main clause must come next:

• Um leið og við komum heim, skrifa ég ...

If you started the sentence with the subject instead, you would say:

• Ég skrifa stutta athugasemd ...

So skrifa ég is normal and expected here.

It can mean either, because for koma, the 1st person plural present and past forms are both komum.

So context has to tell you which one is meant.

In this sentence, it is most naturally understood as present or near-future in meaning:

• when/as soon as we come home
• when/as soon as we get home

In a past narrative, the exact same form could mean we came.

Because heim is a directional adverb meaning home / homeward.

Icelandic often uses it directly after verbs of motion:

• fara heim = go home
• koma heim = come home
• keyra heim = drive home

So you do not need a separate word for to here.

A useful contrast is:

• heim = homeward, to home
• heima = at home

Because athugasemd is the direct object of skrifa, and direct objects are often in the accusative.

The adjective must agree with the noun in gender, number, and case:

• athugasemd = feminine singular noun
• accusative singular = athugasemd
• stuttur in feminine accusative singular = stutta

So:

• stutta athugasemd = a short note/comment

In this sentence, um means about / concerning.

The preposition um normally takes the accusative, so the nouns after it appear in accusative forms:

• lagið = the song
• regnbogann = the rainbow

So:

• um lagið og regnbogann = about the song and the rainbow

Because Icelandic definite endings depend on the noun’s gender and declension type.

Both nouns are definite singular, but they are different kinds of nouns:

• lag is neuter, so the definite form is lagið
• regnbogi is masculine, and its accusative definite singular is regnbogann

So the different endings are grammatical endings, not a difference in meaning.

With í, Icelandic often distinguishes between:

• accusative = movement or direction into
• dative = location in

Here the idea is that the note is being written into the diary, so accusative is natural:

• í dagbókina mína

If you were describing where something already is, dative would be used:

• í dagbókinni minni = in my diary

So the accusative here reflects the idea of putting the writing into the diary.

That is a very common Icelandic pattern. Possessives like minn often come after the noun, especially when the noun is definite.

For example:

• dagbókin mín = my diary
• dagbókina mína = my diary in the accusative

The possessive has to agree with the noun in gender, number, and case. Since dagbókina is feminine singular accusative, the possessive becomes mína.

It has two main parts:

1. Um leið og við komum heim
   This is a subordinate time clause.

2. skrifa ég stutta athugasemd um lagið og regnbogann í dagbókina mína
   This is the main clause.

So the pattern is:

• time clause first
• main clause second

Because the time clause comes first, the main clause shows normal Icelandic V2 word order, which is why you get skrifa ég instead of ég skrifa.