Lying is attention seeking behaviour, her huge outrageous lie was a cry for love, she wanted your undivided attention, negative attention is better than none....she is craving your attention!!
A Dr cannot help, hospitalisation will not help, it will bandaid it...it is not medical, it is psychological, she needs cognitive therapy and so do her parents, therapy is needed to support her and get her to function in an appropriate manner. The therapy sessions will help you as a family unit to ascertain why she does what she does, and how you can 'turn' it around...therapy will give you tools to manage this. Be there and fight tooth and nail for your daughter.
I would say that your daughter have a mental problem, but before or even after she is hospitalized, I would take her to a holistic doctor to evaluate her and treat her naturaly. She is too young to start meds for the rest of her life. I don't know exactly her problem, but the wrong food such as sugars, process food, gluten, caffeine and meat can make her mental illness worse. Do a reasearch about orthomolecular medicine, homeopathy, pranic healing for mental problems and food for healing your body. Also she will need therapy. The less meds she takes, the better. Be strong mom. Love your daughter and accept her. Alway forgive her, but discipline her. Accept your mission in this earth and be brave. God bless you.
How old is your son? Did your daughter give specific details about what she claims he did to her? If she is lying, where would she learn the things to say to make it believable? Watching tv she shouldn't be, unmonitored internet access or a friend who is a terrible influence?
I unfortunately agree it's best for your son not to be living in the same home. There's the saying "Once accused, always a suspect" which doesn't only apply to the police departments opinions. Your son needs to be protected from any false accusations.
About how long has this behavior been going on?
It seems like she's very possessive of you and wants you 100% to herself. Perhaps she's acting out to get more attention from you. Which would explain why she is hateful to your husband and jealous. When your son was talking about moving back in, the second accusation happened...she doesn't want him there to take attention away from her.
Another possibility is that she has been abused by someone else. Oftentimes young victims will accuse someone they know because the actual abuser is frightening or has threatened her in some way. By blaming someone that she knows won't hurt her, she's able to tell you what happened without fear of retaliation from her abuser. Also, not all sexual abuse is visible upon medical examination.
If she has been abused then the change in personality, acting out and lying would be expected effects from emotional trauma.
In my opinion, having a 10 year old committed seems really extreme (unless they are a danger to themselves). Does her psychiatrist agree with the case worker?
I guess my opinion/what advice I'd give you really depends a lot on the time frame that this has all happened. When did she start stealing and lying? How and when did her tantrums turn from regular fits all young children have to being a worrisome behavior? How is she doing in school now compared to years before? Does she have school or neighborhood friends, has she continued the same type of friendships thru this emotional spiral or has she recently gotten new friends?
This is all speculation of course as I don't know your family. I'm just going off of what you've said so far and comparing it to my experience and knowledge of pre-adolescent childhood behavior.