I didn't see your post earlier. I saw it tonight because MamaSource posted your "what happened."
I agree that alone time is important and should help your daughter feel more secure and loved.
I think that letting her express her feelings and telling her that whatever she is feeling is OK is just as important. My 8 yo granddaughter is jealous of her little brother, age 5, in part because he has speech difficulty and may be on the broad spectrum autism disorder. He naturally gets more attention than she does. And he's angry and often hits her. They've developed a hit back sort of routine. Their mother just tells them to knock it off and frequently defends her son by telling her daughter that she started it. I've been there and seen it the other way around or to be more mutual.
When my granddaughter and I are alone we talk about how angry both she and her brother are. I tell her it's OK to be angry but not to hit. suggest over and over for her to use words. Say, "I'm so very angry!" And she does tell me that but it doesn't work so well with her brother because he doesn't have understandable words to deal with his anger.
My granddaughter has always talked non-stop. I do think this is a way of getting attention but I think it's more about her having so many interests and ideas in her head and she needs to talk to process them. She asks a lot of questions and makes a lot adament statements. I see this as her way of feeling like she has some power in her life.
Back to your situation. Screaming and yelling is a way of expressing her anger. I'm a screamer even tho I've worked at stopping it for years and years. At my daughter's house she tells which ever child that is screaming that she doesn't want to hear it, that she knows that they're angry but they have to go to their room until they can be calm again. The child doesn't get punished for screaming. After the child has calmed down my daughter then talks about what had angered them so much and suggests that they can come to her before they get that angry. Unfortunately my daughter isn't able to follow thru on the last statement. She is a single mother working full time and trying to have an adult social life.
When I'm with them, I listen to their exchange and try to intervene before the anger gets so intense. Sometimes I just get each one playing by themselves. Most of the time I've given up on trying to mediate a solution. I do tell each child that I can see that they're angry; therefore it's time to do something else. If they continue to fight over a toy I put the toy up so that neither one plays with it.
My focus is reassuring both children that it's OK to feel angry, frustrated, sad etc. and I give them an opportunity to talk about it. I usually do this separately because my grandson is chronically frustrated because he can't clearly talk about it. With him I listen and even tho I don't understand what he's said I guess at how he's feeling and ask him if that's right. He either nods his head or shakes it. If he shakes it I guess another feeling. We either find out what's bugging him or he loses interest and goes back to playing by himself. I hold both children while we have this talk. With my granddaughter that means waiting until she is calm.
In summary, if one child is angry at the other I separate them, hopefully before the anger gets out of control. I redirct their activities or I send them to seperate rooms. When the anger results in hitting, kicking, throwing before I'm aware I physically step in between them. When it's in the car I pull over to the side and tell them we can't continue until things are back in control. I often hand each one of them something to do. I've learned that playing a CD by the Back Yardigans can keep them calmer. Sometimes my grandson points to the CD's and I put it on. He's learning another way to handle his anger. The goal is to accept their feelings and help them learn new ways of dealing with their anger.
Since the younger child is one you'll need to be more basic with him. Seperation and redirecting will most likely be all that you can do for him. You can try talking with your daughter when it's just the two of you and both of you are calm. Make it a friendly and compassionate conversation about your understanding her anger and finding ways to handle it differently.
There may be other reasons for her jealousy than his actual presence. My grandchildren have several issues. One is his inability to communicated effectively with words while his sister is a talker. Another is that he has a father with whom he visits weekly. Her father is in AZ. She sees that she has to do school work and even tho he's in preschool he doesn't. The two of them have entirely different lives even tho they live together. I think this would be the case with a 6 yo having a 1 yo brother. Talking about the differences helps. Praising her for all that she can do and expressing sympathy that her little brother isn't old enough yet to do them should help.
When we tell my granddaughter she is the big sister she gets angry and frequently says she doesn't want to be the big sister. Just last year, at age 6 and 7, she curled up on my lap and sucked on her finger just as if she were a baby. I'd rock her and tell her she was my baby girl and always would be. Her mother would tell her to act her age which always back fires. She is feeling overwhelmed by her age. She needs times when she can be "a little baby" free from the responsibilities of being a big sister.
I adopted my daughter when she was 7 and she went thru this phase of wanting to be like a baby even tho she was my only child. She had a little brother who was placed in a different home. This illustrates how important it is to accept however our children are feeling and allow them to act their feelings out whenever it's possible.
This also means that we are allowed our own feelings and must find ways of working them out with another adult. It's natural to get frustrated ourselves. I give myself time outs from time to time. And talk my feelings over with friends. I also tell my grandchildren when I'm frustrated and approaching anger which means I'll start yelling. My granddaughter is finally learning sometimes to stop her antics when I say this.
I'm learning many things as a grandmother that I wished I'd known as a Mom. This site is wonderful in teaching me new lessons. I hope that my insights in my situation will help you find a way that works for you.