Is a sentence like
He asked if I would lend him my car.
correct, or should it be
He asked if I lent him my car.
or something else?
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The conditional can definitely be used after "if". See below, for further examples:
He asked if I might be free on Wednesday night .
He asked if it was possible for the dog to run around at the park.
She asked if there was any ice-cream available.
Both of your examples are correct, they just mean different things.
The first is asking if the person is willing to lend the car, and this is used when requesting for something for a time later on e.g. I need a car on Wednesday. Today is Monday, and I don't need the car yet. But I can ask anyway, so I would have to use "would".
The second is asking if the person has already lent the car, that is, does the person who requested have the thing requested? e.g. He asked if I lent him my car. So, did you give your car to him or did you not? This request acts in the instant, and doesn't refer to a future time.
The conditional mood is used after if in indirect questions; if is also used together a modal verb to express a polite request.
She asked if we would like some coffee.
If I could trouble you for your names?
If you wouldn't mind giving him a message?
It is also used to express disappointment.
If I could just be left alone.