If he had known she had too many commitments, he would have done something about it. (Backshifting within the scope of a modally remote preterite)

Question

If he knew she had too many commitments, he would do something about
it.
If he had known he had too many commitments, he would have done
something about it. (This sentence is made up by myself from the above sentence taken from CaGEL)

What would be the grammatical reasoning behind the second "had" (rather than "had had") in the second sentence ("she had too many") taking a past form (if correct)?
Cambridge Grammar of the English Language, p. 152:

There are two reasons... why we need to recognize a backshifted preterite as distinct from an ordinary preterite.
(a) Backshifting within the scope of a modally remote preterite
[11]
i  If he knew she had too many commitments, he would do
something about it.
ii I wish he realised that she had too many commitments.
Example [i] is a remote conditional, with the preterite in knew
expressing modal remoteness, not past time: the time of knowing is
present. So too (certainly in the salient interpretation, the one we
are concerned with here) is the time of her having too many
commitments: it is a matter of knowing in the present about a
situation obtaining in the present. And the same applies to the wish
construction [ii]. In both, therefore, we have the preterite form
had even though there is no reference at all to past time: this cannot represent the ordinary use of the preterite. Nor would it be
valid to subsume the preterite in had in these examples under the
modal remoteness use which applies to the matrix clause preterite
verbs knew and realized. For clearly the unlikelihood in [i] and
the counterfactuality in [ii] apply to his knowing/realizing but not
to her having too many commitments. There is no modal remoteness
meaning attaching to had, for, certainly in [ii], you will in fact
infer that she does have too many commitments. From a semantic point
of view, therefore, the preterite carried by have must be
distinguished from that carried by know or realize, and this
distinction is reflected in the grammar in that irrealis were is not
suitable for was in this construction: If he knew she was/*were too
busy, he would do something about it; I wish he realized she was/*were
too busy.

Anton · Answer

This is an interesting example of how grammatical analysis and formalism may create a line of reasoning that is more obscure and more complex than the prototype.
The Cambridge account is akin to my describing why we draw our hand back when catching a cricket ball, by using all the specialist terms of energy, force, impact, momentum, friction and interatomic interactions. The simple answer is that it hurts if we don’t.
So it is with this question. We may seek understanding of the subtle combination of the rigorous grammatical concepts or we may seek a simple working view of the two sentence alternatives that you consider. Two rather different problems. I am not competent to undertake the former but view the latter as:
If he knew {at this present time} she had {during the time leading up to and including the present time} too many commitments, he would {at this present time} do something about it.
If he had known {at some previous time} she had had {during the period leading up to and including that time} too many commitments, he would {at that time} have done something about it.
This answer is not to dismiss the importance of grammatical analysis but to distinguish it from the more common pursuit of meaning.

If he had known she had too many commitments, he would have done something about it. (Backshifting within the scope of a modally remote preterite)

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