I would use the commas. Here are some suggestions. In British English, the metal cover over the engine of a car is called a "bonnet", not a "hood", which is American English.
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navitasan1-In British English the term "bonnet" is used for the part of the car that covers the engine and not "hood" as in American English.In British English, the term "bonnet", not "hood" as is in American English,
2-In British English the term "bonnet" is used for the part of the car that covers the engine and not "hood" , as in American English.
navitasan… but are the original ones with commas correct or not?The second one could pass as grammatically correct, but not natural, in my opinion. The first would be correct if you changed the second "the" to "a": "the part of a car". This is still not very good because the following "that" could be momentarily mistaken to refer to "car".