Tryggingafélagið borgar bæturnar að því gefnu að bótaskyldan sé skýr.

It is a compound noun meaning the insurance company.

It breaks down like this:

• trygging = insurance
• félag = company, association
• -ið = the suffixed definite article

In compounds, the first part often appears in a linking form, here trygginga-, so tryggingafélag means insurance company, and tryggingafélagið means the insurance company.

Because Icelandic usually puts the at the end of the noun as a suffix instead of using a separate word.

So:

• tryggingafélag = an insurance company
• tryggingafélagið = the insurance company

Here -ið is the definite article for a neuter singular noun in this form.

Borgar is the 3rd person singular present tense of borga, meaning to pay.

Since the subject is Tryggingafélagið = the insurance company, which is third person singular, the verb must also be singular:

• ég borga = I pay
• þú borgar = you pay
• hann/hún/það borgar = he/she/it pays

So Tryggingafélagið borgar means the insurance company pays.

Bæturnar means the compensation, the damages, or the benefits, depending on context.

It comes from:

• bætur = compensation/damages/benefits
• -nar = the definite ending

A useful thing to know is that bætur is very often used in the plural in Icelandic, much like English legal damages is plural even when the meaning feels singular.

So even if English might say the compensation, Icelandic naturally says bæturnar.

It is the direct object of borgar, so it is in the accusative.

In this sentence:

• Tryggingafélagið = subject
• borgar = verb
• bæturnar = object

So the company pays the compensation, and the thing being paid is the object.

With this noun, the nominative and accusative plural definite forms look the same, so the role in the sentence tells you the case.

Yes. Að því gefnu að is a fixed expression meaning:

• provided that
• on condition that
• given that

It is best learned as a whole phrase.

Literally, it is something like with that given, but you should not translate it word-for-word in normal use. In real English, provided that is the best match here.

Because they are doing two different jobs.

• The first að is part of the fixed phrase að því gefnu
• The second að introduces the subordinate clause

So the structure is:

• að því gefnu = given/provided
• að bótaskyldan sé skýr = that the liability is clear

This is similar to English given that.

They are part of the grammar of the fixed expression.

• því is the dative form of það
• gefnu is the dative singular neuter form of the participle/adjective gefinn = given

They agree with each other grammatically. So inside this expression, you get því gefnu.

Even though this can be analyzed grammatically, learners usually benefit most from treating að því gefnu að as one chunk meaning provided that.

Because sé is the present subjunctive of vera = to be.

After expressions like að því gefnu að, Icelandic often uses the subjunctive because the clause expresses a condition, assumption, or requirement, not a plain stated fact.

So:

• er = is, indicative
• sé = be/is, subjunctive

In this sentence, sé is the expected formal and idiomatic choice.

Bótaskyldan is another compound noun. It means something like:

• the liability to pay compensation
• the obligation to compensate

It breaks down as:

• bóta- = compensation, damages
• skylda = duty, obligation, liability
• -n = the definite article

So bótaskylda is liability to compensate, and bótaskyldan is the liability to compensate.

Because it is the subject of the subordinate clause:

• að bótaskyldan sé skýr

Inside that clause:

• bótaskyldan = subject
• sé = verb
• skýr = predicate adjective

Subjects are in the nominative, so bótaskyldan is nominative singular.

Because skýr is a predicate adjective after vera, and predicate adjectives are normally in the nominative and agree with the subject.

Here it agrees with bótaskyldan, which is:

• feminine
• singular
• nominative

So skýr means clear and matches the subject grammatically.

An important point: even though bótaskyldan is definite, predicate adjectives in Icelandic usually use the strong form, not the weak form. So bótaskyldan er skýr is correct.

Yes. It sounds quite formal and legal/insurance-like.

That is because of words and phrases such as:

• Tryggingafélagið
• bæturnar
• að því gefnu að
• bótaskyldan
• sé skýr

This is the kind of wording you would expect in:

• insurance documents
• contracts
• legal writing
• official explanations

So it is perfectly natural Icelandic, but not especially casual everyday conversation.

Yes. The conditional part can come first:

Að því gefnu að bótaskyldan sé skýr, borgar tryggingafélagið bæturnar.

That means the same thing.

Notice that when the sentence starts with the conditional phrase, Icelandic still follows the verb-second pattern in the main clause, so borgar comes before tryggingafélagið.