At first glance, the idea sounds absurd: how can two blond Lannisters be the bastards of the Mad King? But if you dig a little deeper, the theory about Cersei and Jaime's relationship with Aerys II Targaryen suddenly begins to form a complete picture.
Moreover, it adds a new tragic line to the history of the Lannister family and the Game of Thrones, and explains some of the characters' psychological problems.
Fact one: the Mad King had a motive and an opportunity
Aerys II was obsessed with Joanna Lannister, Tywin's wife and cousin. In the book it is directly stated: at her wedding, he allowed himself "liberties", speaking about the right of the first night.
It is not surprising that after some time, Joanna was removed from the court, and a quarrel occurred between Tywin and Aerys. And although she left King's Landing, one day the heroine returned to see her husband - just at the time when she could have conceived twins. This, of course, is not proof, but it is a very suspicious coincidence.

Fact two: the appearance and behavior of the Lannisters
An argument against is the golden hair of Jaime and Cersei. But Joanna was a Lannister by blood, and Targaryens were not always born with silver on their heads - remember Alysanne with her honey hair. Even Jon Snow, being the son of a Targaryen, looks like a typical Stark.
Even Rhaenyra's children from House of the Dragon were born brunettes, so Cersei and Jaime could well have inherited the traits of the mother.
Fact three: their behavior points to the Targaryen curse
Incest is a family matter, as they say in the house of dragons. Brother and sister, keeping their relationship a secret and dreaming of ruling together - a plot in the spirit of an ancient Valyrian family.
Cersei at some point literally repeats the fate of Aerys, destroying enemies with fire, and Jaime, on the contrary, strives for redemption.
"When a Targaryen is born, the gods toss a coin" - this phrase can be applied to them literally.
One becomes a hero, the other a despot.
Fact number four: moral irony
The most elegant thing is in the consequences. If the theory is true, then Tywin's only true son is Tyrion. The one he hated, who, ironically, turned out to be the smartest and most far-sighted. This is not just beautiful - it is in the spirit of Game of Thrones itself.
And Jaime, as a true Targaryen, in this case became not only a regicide, but also a parricide. Tywin would take this secret to his grave. However, many on the Internet believed that Tyrion was one of the Mad King's bastards, but that is a completely different theory.