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I was really scratching my head until I go to the portion that I highlighted in bold.
While I cannot see your dog or situation with my own eyes, the information you provided there says a whole lot. Raising two puppies together has a lot of downfalls. Often-- and I think this is EXACTLY where your problem comes from-- you will see one puppy that will bond with the human portion of the pack and the other puppy will bond mostly with the other puppy. OR it can be doubly difficult where both dogs don't bond with the family and only rely ON EACHOTHER instead of the rest of the pack. This is why I always warn people who tell me they want two pups at once.
Lizzies problem seems to lie in that she has never had the real chance to bond with you, she doesn't get her needs met from you. So, in her mind, she has no NEED or DESIRE to really engage or respond to you once she is bored with the training.
On another level, she most definately has not LEARNED to respond to you through that bond that many of us experience with our dogs. She lives outside, with her brother. To her that is all the pack she needs, and more importantly is all the pack she KNOWS and understand how to respond to.
Her behavior towards you isn't stubborness or even boredom. Its a lack of any pack infastructure that would otherwise naturally assist her training, the desire to please and so on. Lizzie has no idea how to engage with you beyond what she knows of you playing with and feeding her. She has no need to because she has what she needs: her brother as her pack and what she is comfortable with.
If you want a relationship with your dogs I would start by seperating them, start building a relationship with them, make them part of your family. Without building a pack relationship with her she will not see the reason she should engage you in training or otherwise. You can train her on a shallow level and have her respond to you but the training won't stick unless you dig deeper and give her what she needs. In return she will give you what you give--repsonsive relationship.