The sentence is carelessly constructed, so it won't sustain a strict parse; but here's what's going on.

Over here is attached to contention. Verbs like contend, quarrel, dispute, argue often take a preposition phrase with over as a complement designating the matter which is disputed:

The sisters quarreled over who should get their mother's silver.
The king and his opponents contended over the rights of minorities and parlements.

A noun derived from a verb often may take as a post-positioned modifier the same sort of preposition phrase which acts as the verb's complement.

[_CLAUSEThey contended over X.] → [_{NOUN PHRASE}Their contention over X]

From there the attributive PP is easily recast as a predicative PP.

[_{NOUN PHRASE}Their contention over X] → [_CLAUSETheir contention was over X.]

And that finite clause may be recast as a non-finite clause so it can be subordinated:

[_{FINITE CLAUSE}Their contention was over X.] → [_{NON-FINITE CLAUSE}... their contention being over X.]

A participle clause like this may be employed as an absolute construction:

... a grammatical construction standing apart from a normal or usual syntactical relation with other words or sentence elements. It can be a non-finite clause that is subordinate in form and modifies an entire sentence ... —Wikipedia.

The participle clause headed by being modifies the main clause, supplying the matters over which the king's authority was challenged.

What throws a monkey wrench into this particular absolute construction is Dr. Linton's expansion of contention to sources of contention.

The main sources of contention being over X ...

Now the subject of being is no longer contention but sources; and it is not idiomatic to speak of a source as over X. The sources of contention, in the sense in which Dr. Linton uses source, is X itself. Dr. Linton should have written:

... the main sources of contention being over the rights of religious minorities (Jansenists and Protestants) and the rights of the parlements.

I hasten to add that this a very minor error, which would pass entirely unnoticed in speech. Dr. Linton is a graceful writer; I imagine she rewrote this sentence many times, and somehow the over never got deleted. It's very easy to make this sort of error, and very hard to spot it.

The sentence is carelessly constructed, so it won't sustain a strict parse; but here's what's going on.

Over here is attached to contention. Verbs like contend, quarrel, dispute, argue often take a preposition phrase with over as a complement designating the matter which is disputed:

The sisters quarreled over who should get their mother's silver.
The king and his opponents contended over the rights of minorities and parlements.

A noun derived from a verb often may take as a post-positioned modifier the same sort of preposition phrase which acts as the verb's complement.

[_CLAUSEThey contended over X.] → [_{NOUN PHRASE}Their contention over X]

From there the attributive PP is easily recast as a predicative PP.

[_{NOUN PHRASE}Their contention over X] → [_CLAUSETheir contention was over X.]

And that finite clause may be recast as a non-finite clause so it can be subordinated:

[_{FINITE CLAUSE}Their contention was over X.] → [_{NON-FINITE CLAUSE}... their contention being over X.]

A participle clause like this may be employed as an absolute construction:

... a grammatical construction standing apart from a normal or usual syntactical relation with other words or sentence elements. It can be a non-finite clause that is subordinate in form and modifies an entire sentence ... —Wikipedia.

The participle clause headed by being modifies the main clause, supplying the matters over which the king's authority was challenged.

What throws a monkey wrench into this particular absolute construction is Dr. Linton's expansion of contention to sources of contention.

The main sources of contention being over X ...

Now the subject of being is no longer contention but sources; and it is not idiomatic to speak of a source as over X. The sources of contention, in the sense in which Dr. Linton uses source, is X itself. Dr. Linton should have written:

... the main sources of contention being over the rights of religious minorities (Jansenists and Protestants) and the rights of the parlements.

I hasten to add that this is a very minor error, which would pass entirely unnoticed in speech. Dr. Linton is a graceful writer; I imagine she rewrote this sentence many times, and somehow the over never got deleted. It's very easy to make this sort of error, and very hard to spot it.

The sentence is carelessly constructed, so it won't sustain a strict parse; but here's what's going on.

Over here is attached to contention. Verbs like contend, quarrel, dispute, argue often take a preposition phrase with over as a complement designating the matter which is disputed:

The sisters quarreled over who should get their mother's silver.
The king and his opponents contended over the rights of minorities and parlements.

A noun derived from a verb often may take as a post-positioned modifier the same sort of preposition phrase which acts as the verb's complement.

[_CLAUSEThey contended over X.] → [_{NOUN PHRASE}Their contention over X]

From there the attributive PP is easily recast as a predicative PP.

[_{NOUN PHRASE}Their contention over X] → [_CLAUSETheir contention was over X.]

And that finite clause may be recast as a non-finite clause so it can be subordinated:

[_{FINITE CLAUSE}Their contention was over X.] → [_{NON-FINITE CLAUSE}... their contention being over X.]

A participle clause like this may be employed as an absolute construction:

... a grammatical construction standing apart from a normal or usual syntactical relation with other words or sentence elements. It can be a non-finite clause that is subordinate in form and modifies an entire sentence ... —Wikipedia.

The participle clause headed by being modifies the main clause, supplying the matters over which the king's authority was challenged.

What throws a monkey wrench into this particular absolute construction is Dr. Linton's expansion of contention to sources of contention.

The main sources of contention being over X ...

Now the subject of being is no longer contention but sources; and it is not idiomatic to speak of a source as over X. X itself is the source of contention. Dr. Linton should have written:

... the main sources of contention being over the rights of religious minorities (Jansenists and Protestants) and the rights of the parlements.

I hasten to add that this a very minor error, which would pass entirely unnoticed in speech. Dr. Linton is a graceful writer; I imagine she rewrote this sentence many times, and somehow the over never got deleted. It's very easy to make this sort of error, and very hard to spot it.

The sentence is carelessly constructed, so it won't sustain a strict parse; but here's what's going on.

Over here is attached to contention. Verbs like contend, quarrel, dispute, argue often take a preposition phrase with over as a complement designating the matter which is disputed:

The sisters quarreled over who should get their mother's silver.
The king and his opponents contended over the rights of minorities and parlements.

A noun derived from a verb often may take as a post-positioned modifier the same sort of preposition phrase which acts as the verb's complement.

[_CLAUSEThey contended over X.] → [_{NOUN PHRASE}Their contention over X]

From there the attributive PP is easily recast as a predicative PP.

[_{NOUN PHRASE}Their contention over X] → [_CLAUSETheir contention was over X.]

And that finite clause may be recast as a non-finite clause so it can be subordinated:

[_{FINITE CLAUSE}Their contention was over X.] → [_{NON-FINITE CLAUSE}... their contention being over X.]

A participle clause like this may be employed as an absolute construction:

... a grammatical construction standing apart from a normal or usual syntactical relation with other words or sentence elements. It can be a non-finite clause that is subordinate in form and modifies an entire sentence ... —Wikipedia.

The participle clause headed by being modifies the main clause, supplying the matters over which the king's authority was challenged.

What throws a monkey wrench into this particular absolute construction is Dr. Linton's expansion of contention to sources of contention.

The main sources of contention being over X ...

Now the subject of being is no longer contention but sources; and it is not idiomatic to speak of a source as over X. The sources of contention, in the sense in which Dr. Linton uses source, is X itself. Dr. Linton should have written:

... the main sources of contention being over the rights of religious minorities (Jansenists and Protestants) and the rights of the parlements.

I hasten to add that this a very minor error, which would pass entirely unnoticed in speech. Dr. Linton is a graceful writer; I imagine she rewrote this sentence many times, and somehow the over never got deleted. It's very easy to make this sort of error, and very hard to spot it.