Why don't the personal possessive adjectives его (his), её (her), and их (their) decline for the dative case?

Question

Let's use a simple sentence to illustrate this further:

[SUBJECT NOUN] gave [PERSONAL POSSESSIVE] [INDIRECT OBJECT] a [DIRECT OBJECT].

If I want to say:

He gave my brother a book.
He gave your brother a book.
He gave our brother a book.

I use the dative declension for my, your, or our.

Он дал моему брату книгу.
Он дал твоему брату книгу.
Он дал нашему брату книгу.

But if, instead, I want to say:

He gave her brother a book.
He gave their brother a book.

I just use его or их:

Он дал её брату книгу.
Он дал их брату книгу.

Why don't these forms decline like the others?

As for research previously conducted, I can barely even find a web page that makes reference to this fact, let alone explain why this is so. The page below from "Learn Russian Step by Step," was probably the most straightforward about just mentioning it:

Possessive pronouns in the Dative case

Answer 1

I know it is an unsatisfying answer, but it just happened to be this way.

For a fairly long time, Slavic languages didn't use to have a personal pronoun for the 3rd person. Instead, they used the demonstratives *и, *ꙗ, and *ѥ, which is where the forms like её and его come from.

Pretty early on, the nominative forms of these pronouns got replaced (suppletion) by the nominative forms of another set of demonstratives: онъ, она, оно.

As time went on, these combined forms became personal pronouns for the 3rd person, which, funnily enough, resulted in the formation of a completely new system of demonstratives like тот and этот.

Now, as you may expect, possession also used to have only 1st and 2nd person pronouns. If you wanted to refer to the possession by a third person, you would simply use an adjective like сестрин (sister's) or restructure your entire phrase.

This is why мой and твой decline like that, in a very adjectival way (compare "мой" to "голубой").

With the 3rd person getting its pronouns, the possession also needed to change. But there wasn't really a form to handle it. So the genitive of 3rd person pronoun was used instead, but it can't change with the cases anymore since it has to be in a genitive case to show possession.

So that is why 1st and 2nd person possessives decline differently from 3rd person ones.

However, there is an interesting process that started back in medieval times and was subsequently fought by the Soviet reforms with questionable results. You see, you are not the only person who found the need to have a 3rd person possessive that shows what case it is in.

It seems that for the last 500 years or so, the language tried to fill the void by doing the only obvious thing: grabbing the existing 3rd person possessive and slamming an adjectival ending on it.

All the way back in XVII century, there was already significant use of the form ихний (their). Over the years, you will also see forms like егоный/евоный, еёный/ейный pop up here and there. These are always considered extremely wrong and extremely colloquial by the school system, yet they just keep showing up.

If you are interested in these kinds of topics, I recommend picking up a copy of "Историческая грамматика русского языка" by В. Иванов. (Аванесов Р. И., Иванов В. В. Историческая грамматика русского языка: Морфология, глагол. – Изд-во" Наука,", 1982.)

Answer 2

The point is, его, её, их lack not only a special Dative case form, but they are not declinable at all, they have the same form for all the cases, both singular and plural, for example, in plural:

от его друзей (Genitive pl.) “from his friends”
c его друзьями (Instrumental pl.) “with his friends”
на его друзьях (Prepositional pl.) “on his friends”

The reason for that is that although его, её, их are considered possessive pronouns, in fact and historically they are just the Genitive case forms of the personal pronouns он (оно), она, они. (cf. Я боюсь его (Genitive) “I'm afraid of him”). Attributes in the Genitive case (and the possessive pronouns are attributes) are called несогласованные определения (“uncoordinated / not agreed attributes”) since such attributes always stay in the Genitive case irrespective of which case the head noun is in. For example, in the following sentences Петра “of Peter / Peter's” is a Genitive case attribute:

Nom. брат Петра “a brother of Peter”
Gen. без брата Петра “without a brother of Peter”
Dat. брату Петра “to/for a brother of Peter”
Acc. [вижу] брата Петра “[I see] a brother of Peter”
Instr. с братом Петра “with a brother of Peter”
Prep. о брате Петра “about a brother of Peter”

As you can see, его, её, их never change their form in any case of singular or plural whatsoever since historically they are the Genitive case forms of the corresponding personal pronouns, and not possessive pronouns proper like мой or ваш. Genitive case attributes remain unchanged when the head of their noun phrase changes its case.