syntactic analysis - How to tell if something is a core complement or a non-core complement?

CaGEL on page 216 cite the following:

"Kim gave the key to Pat"

An NP indirectly related to the verb through the preposition is referred as an oblique. The phrase "to Pat" is a non-core compliment of the verb give, but the NP Pat is an oblique.

In a double object construction where both the noun phrase + prepositional phrases are both "core complements" either, is referred to as a direct or indirect object.

These verbs are called double object verbs. When we have two noun phrases after the verb the first noun phrase is the indirect object and the second noun phrase is the direct object.

Are old man and animals in the second row of the picture above, not noun phrases too? Like the oblique NP Pat, in "Kim gave the key to Pat".

My guess is that where there are two possibilities: core complements (NP and NP) and when there is a core complement and non-core complement (NP and PP):
Sentences with two core complements next to each other in the clause-final position; the first component can be seen as an indirect object, and the following NP after, is the direct object.
In a sentence where there is a core complement next to a non-core complement in the clause-final position; only the first complement is considered core which is usually a NP and the direct object, and the second component which is usually the (PP) a non-core complement and not a indirect object.

In most cases, it is easy to distinguish such as below:

Kim gave Pat _______ .

This does not answer what Kim gave Pat, so there must be another core complement that being "the key". Traditional grammar analysis would class "the key" in this case as the direct object, while "Pat" the indirect object. Does CaGEL think this too when there is 2 noun phrases following the verb?

Kim gave the key. [to Pat]

Kim gave Pat _____ .

The direct object in (1) can stand on its own without the non-core "to-complement"; the to-complement is not an indirect object according to CaGEL and the direct object would be "the key".
But the indirect object in (2) cannot stand on its own, so the what-complement has to be answered and is core; the what-complement "the key" is a direct object. Both "Pat" and "the key" in (2) are noun phrases.

Wikipedia considers the CaGEL's example "She gave the key to Pat" as an oblique dative shift, where the prepositional phrase is a non-core complement. They make this distinction:

In the oblique dative (OD) form, the verb takes a noun phrase (NP) and a prepositional phrase (PP), the second of which is not a core argument:

John gave [NP a book] [PP to Mary].

In the double object construction (DOC) form the verb takes two noun phrases, both of which are core arguments:

John gave [NP Mary] [NP a book].

The assumption is that the “oblique objects” which you get through dative alternation with a prepositional phrase are not core arguments

However, in some cases it's harder to distinguish whether a complement is core or not. This means that the PP is not always a non-core complement to the NP. For example, Wikipedia also cites the following examples as DOC, even though there is a dative alternation, with NP and PP as its core argument:

John bought [NP a cake] [PP for Mary]

John acquired [NP a new car] [PP for Mary]

My question is what do you consider as "core" and "non-core complements"?

(The bold is what I believed to be the core complements and the italics the non-core complements)

*John bought Mary a cake.
*John acquired Mary a new car.

? John bought Mary.
? John acquired Mary.

Can the above stand on their own, but in:

"Kim gave Pat."

it cannot?

✓ John bought a cake.
✓ John acquired a new car.

*John bought a cake Mary.
*John acquired a new car Mary.

✓ John bought a cake for Mary.
✓ John acquired a new car for Mary.

There is a preference for the NP (a new car/cake) to be core complements to the verb and the NP Mary to be a non-core complement indirectly related to the verb. But there doesn't seem to be an explanation as to why the NP "a cake" and "a new car" can stand on their own; but the NP Mary and Pat cannot. The correct form would be "for Mary" and it would still have to follow the core complement, it cannot precede the core complement or stand on its own. For example it cannot be:

*John acquired for Mary.
*John bought for Mary.

? John acquired for Mary a new car.
? John bought for Mary a cake.

✓ John acquired a new car for Mary.
✓ John bought a cake for Mary.

But below is possible:

✓ John acquired a new car.
✓ John bought a cake.

John bought a cake.

and

John bought a cake for Mary.

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