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In S4E6, during the trial where Tyrion is accused of regicide, Shae provides a false testimony to the jury. The question of why Shae betrayed Tyrion has already been covered, but I haven’t found anything about Shae betraying Sansa. During the testimony, Shae says things like

SHAE: I know that he's guilty. He and Sansa planned it together.

...

SHAE: After that, all he wanted was her. But she wouldn't let him into her bed. So he promised to kill King Joffrey for her.

However, previously Shae said that she would kill for her (Sansa) and acted in ways of supporting Sansa, like trying to cover up her blood-stained sheets, etc. This testimony would most likely result in a death sentence for Sansa, if she were to be captured (her life was already at stake but this would make it ‘official’). So why implicate Sansa, the child she tried to protect so fiercely? Why not just say Tyrion killed the king and leave Sansa out of it?

I don’t remember how the trial went in the books, so perhaps it’s different there. While a comparison between the show and the books would be great, I’m mainly looking for answers based on the show.

The show appears to have mainly gone down the route of scorned ex-lover and so Shae forgets her protectiveness and friendship to Sansa just for the sake of plot... like so many things with the show.

TYRION: I'm a married man. My wife has suffered a great deal, as you well know. I don't want her to suffer anymore on my account. I need to uphold my vows.

SHAE: She doesn't want you.

[...]

SHAE: We will fight them together. It's like you said, I am yours and you are mine.

TYRION: You're a whore! Sansa is fit to bear my children and you are not. I can't be in love with a whore. I can't have children with a whore. How many men have you been with? 500? 5,000?

SHAE: How many whores have you been with?

TYRION: I have enjoyed my time with all of them and I have enjoyed my time with you most of all. But now that time is over.

TYRION walks over and opens the door. SHAE sobs. BRONN enters.

TYRION: You'll have a comfortable life in Pentos. Bronn will escort you to your ship.

SHAE continues to sob. BRONN clasps her shoulder to lead her out. She turns quickly and slaps him, then storms off, crying. BRONN looks to TYRION, then exits. TYRION walks to the table and violently slaps his cup to the ground.

_{Game of Thrones, Season 4 Episode 2, "The Lion and the Rose"}

And the things she says at the trial again indicate that she is "annoyed" that Tyrion wed Sansa.

SHAE: I am a whore. Remember? That was before he married Sansa. After that, all he wanted was her. But she wouldn't let him into her bed. So he promised to kill King Joffrey for her.

_{Game of Thrones, Season 4 Episode 6, "The Laws of Gods and Men"}

The implication, before the above conversation takes place, also appears to be that Cersei and Tywin have noticed what is going on and intercepted her leaving. They then probably used Tyrion's marriage to Sansa and her being hurt to manipulate her to do what they want.

MACE bows and exits. SHAE brings a tray to the table and sets it before SANSA.

CERSEI: She's the whore I told you about. The dark-haired one.

SHAE exits.

TYWIN: Have her brought to the Tower of the Hand before the wedding.

TYRION looks concerned. PODRICK places a large tome before JOFFREY and TYRION walks to the front of the table.

_{Game of Thrones, Season 4 Episode 2, "The Lion and the Rose"}

This is made even clearer when Shae is found in Tywin's chambers by Tyrion being used as a whore. It seems incredibly likely that Tywin manipulated her to do as he wanted.

INT. TYWIN’S BEDCHAMBER - NIGHT

A dark-haired woman lies atop the bedclothes, her face turned away from us, naked save for the golden chains she wears round her neck.

On a wood board atop the bedside table there’s a halved apple, cheese, and a small, sharp knife.

Staring at the naked woman, not really believing who she is, Tyrion walks quietly toward the bed.

Not quietly enough.

SHAE: Tywin? My lion?

She turns with a smile on her face. The smile disappears when she sees Tyrion.

_{Game of Thrones, Season 4 Episode 10, "The Children" - Official script}

This is backed up even more by the fact that in the books Cersei almost certainly had words with her to persuade her to do as she wants.

She followed them back inside and watched as they bundled the girl up in her father's bloody blankets. Shae, her name was Shae. They had last spoken the night before the dwarf's trial by combat, after that smiling Dornish snake offered to champion him. Shae had been asking about some jewels Tyrion had given her, and certain promises Cersei might have made, a manse in the city and a knight to marry her. The queen made it plain that the whore would have nothing of her until she told them where Sansa Stark had gone. "You were her maid. Do you expect me to believe that you knew nothing of her plans?" she had said. Shae left in tears.

_{A Feast for Crows, Cersei I}

Tywin appears to have convinced her to do it. When Tyrion escapes he finds her in Tywin's bed, and she seems to think that he is in fact Tywin returning from the bathroom. So at that point Shae and Tywin appear to be sleeping together.

The TV show doesn't expand on their relationship. It's not clear what Tywin offered her, perhaps just a comfortable lifestyle as his companion or perhaps something else. But it's clear that Tywin is controlling her to some degree, with her testimony being exchanged for favours.

Thus we can assume that it was Tywin who wanted Sansa implicated. Sansa is a loose end now, with Tyrion exiled to The Wall she needs to be controlled another way. Marriage to Tyrion ensured that any children or other husbands would not pose a threat to House Lannister. Remember how keen Tywin was for Tyrion to produce an heir. Convicting Sansa of regicide and imprisonment/execution would achieve the same thing.

In addition to TheLethalCarrot's excellent answer, I would add that Shae's false testimony is being provided for public consumption. That makes it important that she present a compelling motive for Tyrion to kill his brother.

The viewer knows that Joffrey is a monster, and that's reason enough for anyone to kill him. But that's not the public perception. If she had testified that Tyrion murdered Joffrey just because he felt like it, her testimony would not have been as effective.