Suggest doing something

Yes, the book which I quoted from is British. To wind up, I should go along with what has been said in that book that (in BrE) if someone suggests doing something, that means they are going to be involved in that activity too. 'My colleague suggested sharing a car to work' should be okay, but 'The doctor suggested taking exercise' doesn't sound normally likely. A doctor would rather suggest that someone should do exercise.

AlpheccaStarsYou need to contrast the speech patterns using pronouns with that of nouns.

I read somewhere, 'We suggest ideas and not suggest to someone'. Also as you suggested, a natural sounding sentence requires a prepositional phrase to go only after a complement ("compliment"). An exception is when you 'contrast the speech patterns using pronouns with that of nouns'. Then, similar to the other example sentence I took, this should be correct:

The doctor suggested to the patient (a noun phrase or a noun) that they (a pronoun) should take exercise. ?

But this should not be correct or natural:

The doctor suggested me (a pronoun) that I (a pronoun again!) should take exercise. ?

Suggesting an idea in this should also be correct:

The doctor suggested I/me taking exercise/exercising. ?

Last but not least in this one of my long-winded comments, a thank-you may have been due to AlpheccaStars mainly, for putting across the answers, also to the anonymous one saving me a word 'taking', and to Chris for making it even more concise for me further saving me an 'ing' from 'exercising'!